यदि \(^{n}C_{3}=4,^{n}C_{2}\) है, तो (n) का मान क्या है?
If \(^{n}C_{3}=4,^{n}C_{2}\), what is the value of (n)?
#combinations
#ncr
#identity
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (13)
B (14)
C (15)
D (16)
Explanation opens after your attempt
Step 1
Concept
The ratio is \(\frac{^{n}C_{3}}{^{n}C_{2}}=\frac{n-2}{3}\), so (n=14). In exams write the ratio first.
Step 2
Why this answer is correct
The correct answer is B. (14). The ratio is \(\frac{^{n}C_{3}}{^{n}C_{2}}=\frac{n-2}{3}\), so (n=14). In exams write the ratio first.
Step 3
Exam Tip
अनुपात \(\frac{^{n}C_{3}}{^{n}C_{2}}=\frac{n-2}{3}\) होगा, इसलिए (n=14)। परीक्षा में पहले अनुपात लिखें।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{4}=^{n}C_{6}\) है, तो (n) का मान क्या है?
If \(^{n}C_{4}=^{n}C_{6}\), what is the value of (n)?
#combinations
#symmetry
#ncr
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
For same (n), \(^{n}C_{r}=^{n}C_{s}\) implies (r=s) or (r+s=n). Here (n=10).
Step 2
Why this answer is correct
The correct answer is C. (10). For same (n), \(^{n}C_{r}=^{n}C_{s}\) implies (r=s) or (r+s=n). Here (n=10).
Step 3
Exam Tip
समान (n) के लिए \(^{n}C_{r}=^{n}C_{s}\) होने पर (r=s) या (r+s=n)। यहां (n=10) है।
Login to save your score, XP, coins and progress. Login
यदि \(^{12}C_{r}=^{12}C_{r+4}\) है, तो (r) का मान क्या है?
If \(^{12}C_{r}=^{12}C_{r+4}\), what is the value of (r)?
#combinations
#ncr
#equation
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
In the equality, (r+(r+4)=12). Hence (r=4).
Step 2
Why this answer is correct
The correct answer is B. (4). In the equality, (r+(r+4)=12). Hence (r=4).
Step 3
Exam Tip
समानता में (r+(r+4)=12) होगा। इससे (r=4) मिलता है।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{2}=66\) है, तो \(^{n}C_{4}\) का मान क्या होगा?
If \(^{n}C_{2}=66\), what is the value of \(^{n}C_{4}\)?
#combinations
#ncr
#advanced
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (495)
B (220)
C (330)
D (715)
Explanation opens after your attempt
Step 1
Concept
From \(^{n}C_{2}=66\), (n=12). Therefore \(^{12}C_{4}=495\).
Step 2
Why this answer is correct
The correct answer is A. (495). From \(^{n}C_{2}=66\), (n=12). Therefore \(^{12}C_{4}=495\).
Step 3
Exam Tip
\(^{n}C_{2}=66\) से (n=12) मिलता है। इसलिए \(^{12}C_{4}=495\)।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{5}:^{n}C_{4}=7:2\) है, तो (n) का मान क्या है?
If \(^{n}C_{5}:^{n}C_{4}=7:2\), what is the value of (n)?
#combinations
#ratio
#trap
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
\(\frac{^{n}C_{5}}{^{n}C_{4}}=\frac{n-4}{5}=\frac{7}{2}\) gives no integer (n). Always check validity of the ratio.
Step 2
Why this answer is correct
The correct answer is D. (21). \(\frac{^{n}C_{5}}{^{n}C_{4}}=\frac{n-4}{5}=\frac{7}{2}\) gives no integer (n). Always check validity of the ratio.
Step 3
Exam Tip
\(\frac{^{n}C_{5}}{^{n}C_{4}}=\frac{n-4}{5}=\frac{7}{2}\) से (n=21.5) नहीं आता, इसलिए सही अनुपात जांचना जरूरी है। यहां कोई पूर्णांक (n) संभव नहीं है।
Login to save your score, XP, coins and progress. Login
(12) विद्यार्थियों में से (5) की समिति बनानी है, लेकिन दो विशेष विद्यार्थी साथ-साथ ही चुने जा सकते हैं। कितनी समितियां बनेंगी?
A committee of (5) is to be formed from (12) students, but two particular students can be selected only together. How many committees are possible?
#committee
#restriction
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (252)
B (330)
C (372)
D (420)
Explanation opens after your attempt
Step 1
Concept
If both special students are selected, count \(^{10}C_{3}\); if both are not selected, count \(^{10}C_{5}\). Total (120+252=372).
Step 2
Why this answer is correct
The correct answer is C. (372). If both special students are selected, count \(^{10}C_{3}\); if both are not selected, count \(^{10}C_{5}\). Total (120+252=372).
Step 3
Exam Tip
दोनों विशेष चुने जाएं तो \(^{10}C_{3}\), और दोनों न चुने जाएं तो \(^{10}C_{5}\)। कुल (120+252=372)।
Login to save your score, XP, coins and progress. Login
(8) पुरुष और (6) महिलाओं में से (5) सदस्यों की समिति बनानी है जिसमें कम से कम (3) महिलाएं हों। कितने चयन संभव हैं?
From (8) men and (6) women, a committee of (5) is to be formed with at least (3) women. How many selections are possible?
#committee
#atleast
#gender
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (420)
B (500)
C (560)
D (616)
Explanation opens after your attempt
Step 1
Concept
Cases are (3W2M), (4W1M), and (5W0M). The value is \(^{6}C_{3}{}^{8}C_{2}+^{6}C_{4}{}^{8}C_{1}+^{6}C_{5}=560\).
Step 2
Why this answer is correct
The correct answer is C. (560). Cases are (3W2M), (4W1M), and (5W0M). The value is \(^{6}C_{3}{}^{8}C_{2}+^{6}C_{4}{}^{8}C_{1}+^{6}C_{5}=560\).
Step 3
Exam Tip
मामले (3W2M), (4W1M), (5W0M) हैं। मान \(^{6}C_{3}{}^{8}C_{2}+^{6}C_{4}{}^{8}C_{1}+^{6}C_{5}=560\)।
Login to save your score, XP, coins and progress. Login
(10) खिलाड़ियों में से (4) का समूह चुनना है, लेकिन कप्तान और उपकप्तान दोनों साथ नहीं आ सकते। कितने समूह बनेंगे?
A group of (4) is to be chosen from (10) players, but the captain and vice-captain cannot both be included. How many groups are possible?
#combinations
#exclusion
#committee
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (140)
B (168)
C (182)
D (196)
Explanation opens after your attempt
Step 1
Concept
Total selections are \(^{10}C_{4}=210\). Remove \(^{8}C_{2}=28\) selections containing both special players, giving (182).
Step 2
Why this answer is correct
The correct answer is C. (182). Total selections are \(^{10}C_{4}=210\). Remove \(^{8}C_{2}=28\) selections containing both special players, giving (182).
Step 3
Exam Tip
कुल \(^{10}C_{4}=210\) हैं। दोनों विशेष साथ होने वाले \(^{8}C_{2}=28\) हटाएं, उत्तर (182)।
Login to save your score, XP, coins and progress. Login
(7) अलग पुस्तकों में से (4) पुस्तकें चुननी हैं, पर दो निर्धारित पुस्तकें साथ-साथ नहीं चुनी जा सकतीं। कितने चयन संभव हैं?
From (7) distinct books, (4) books are to be selected, but two specified books cannot be selected together. How many selections are possible?
#books
#restriction
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (20)
B (25)
C (30)
D (35)
Explanation opens after your attempt
Step 1
Concept
Total selections are \(^{7}C_{4}=35\). If both specified books are together, \(^{5}C_{2}=10\) selections are removed.
Step 2
Why this answer is correct
The correct answer is B. (25). Total selections are \(^{7}C_{4}=35\). If both specified books are together, \(^{5}C_{2}=10\) selections are removed.
Step 3
Exam Tip
कुल \(^{7}C_{4}=35\) हैं। दोनों निर्धारित पुस्तकें साथ होने पर \(^{5}C_{2}=10\) चयन हटेंगे।
Login to save your score, XP, coins and progress. Login
(9) बिंदुओं में से कोई (3) सरल रेखा पर नहीं हैं। कितने त्रिभुज बनाए जा सकते हैं?
There are (9) points with no (3) collinear. How many triangles can be formed?
#geometry
#triangles
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (72)
B (81)
C (84)
D (90)
Explanation opens after your attempt
Step 1
Concept
A triangle needs (3) points. Hence the number is \(^{9}C_{3}=84\).
Step 2
Why this answer is correct
The correct answer is C. (84). A triangle needs (3) points. Hence the number is \(^{9}C_{3}=84\).
Step 3
Exam Tip
त्रिभुज के लिए (3) बिंदु चाहिए। इसलिए संख्या \(^{9}C_{3}=84\) है।
Login to save your score, XP, coins and progress. Login
(11) बिंदुओं में से (5) बिंदु एक ही रेखा पर हैं और बाकी में कोई (3) सरल रेखा पर नहीं हैं। कितने त्रिभुज बनेंगे?
Among (11) points, (5) are collinear and no other (3) points are collinear. How many triangles can be formed?
#geometry
#collinear
#triangles
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (155)
B (160)
C (165)
D (170)
Explanation opens after your attempt
Step 1
Concept
Subtract collinear choices \(^{5}C_{3}=10\) from total \(^{11}C_{3}=165\). The answer is (155).
Step 2
Why this answer is correct
The correct answer is A. (155). Subtract collinear choices \(^{5}C_{3}=10\) from total \(^{11}C_{3}=165\). The answer is (155).
Step 3
Exam Tip
कुल \(^{11}C_{3}=165\) से कोलिनियर \(^{5}C_{3}=10\) घटाएं। उत्तर (155) है।
Login to save your score, XP, coins and progress. Login
(10) भुजाओं वाले उत्तल बहुभुज में विकर्णों की संख्या कितनी है?
How many diagonals are there in a convex polygon with (10) sides?
#polygon
#diagonals
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (30)
B (35)
C (40)
D (45)
Explanation opens after your attempt
Step 1
Concept
Total ways to choose two vertices are \(^{10}C_{2}\). Subtract (10) sides to get (35) diagonals.
Step 2
Why this answer is correct
The correct answer is B. (35). Total ways to choose two vertices are \(^{10}C_{2}\). Subtract (10) sides to get (35) diagonals.
Step 3
Exam Tip
दो शीर्ष चुनने के कुल तरीके \(^{10}C_{2}\) हैं। भुजाएं (10) घटाने पर विकर्ण (35) मिलते हैं।
Login to save your score, XP, coins and progress. Login
(12) भुजाओं वाले उत्तल बहुभुज में कितने चतुर्भुज बनाए जा सकते हैं?
How many quadrilaterals can be formed from a convex polygon with (12) sides?
#polygon
#quadrilateral
#ncr
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (420)
B (450)
C (495)
D (520)
Explanation opens after your attempt
Step 1
Concept
Any (4) vertices form one quadrilateral. Thus the number is \(^{12}C_{4}=495\).
Step 2
Why this answer is correct
The correct answer is C. (495). Any (4) vertices form one quadrilateral. Thus the number is \(^{12}C_{4}=495\).
Step 3
Exam Tip
किसी भी (4) शीर्षों से एक चतुर्भुज बनता है। इसलिए संख्या \(^{12}C_{4}=495\) है।
Login to save your score, XP, coins and progress. Login
एक क्रिकेट दल में (7) बल्लेबाज और (6) गेंदबाज हैं। (6) खिलाड़ियों का समूह चुनना है जिसमें कम से कम (2) गेंदबाज हों। कितने तरीके हैं?
A cricket squad has (7) batters and (6) bowlers. A group of (6) players is to be chosen with at least (2) bowlers. How many ways are possible?
#sports
#atleast
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (1589)
B (1603)
C (1624)
D (1716)
Explanation opens after your attempt
Step 1
Concept
Subtract cases with (0) and (1) bowler from \(^{13}C_{6}\). (1716-7-120=1589).
Step 2
Why this answer is correct
The correct answer is A. (1589). Subtract cases with (0) and (1) bowler from \(^{13}C_{6}\). (1716-7-120=1589).
Step 3
Exam Tip
कुल \(^{13}C_{6}\) से (0) और (1) गेंदबाज वाले मामले घटाएं। (1716-7-120=1589)।
Login to save your score, XP, coins and progress. Login
(6) भारतीय और (5) विदेशी लेखकों में से (4) लेखकों का पैनल बनाना है जिसमें दोनों प्रकार के लेखक हों। कितने चयन संभव हैं?
From (6) Indian and (5) foreign authors, a panel of (4) authors is to be formed with both types present. How many selections are possible?
#panel
#inclusion
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (275)
B (300)
C (310)
D (325)
Explanation opens after your attempt
Step 1
Concept
Subtract all-Indian \(^{6}C_{4}=15\) and all-foreign \(^{5}C_{4}=5\) from total \(^{11}C_{4}=330\). Answer (310).
Step 2
Why this answer is correct
The correct answer is C. (310). Subtract all-Indian \(^{6}C_{4}=15\) and all-foreign \(^{5}C_{4}=5\) from total \(^{11}C_{4}=330\). Answer (310).
Step 3
Exam Tip
कुल \(^{11}C_{4}=330\) से केवल भारतीय \(^{6}C_{4}=15\) और केवल विदेशी \(^{5}C_{4}=5\) हटाएं। उत्तर (310)।
Login to save your score, XP, coins and progress. Login
(15) प्रश्नों में से (10) प्रश्न हल करने हैं। पहले (5) प्रश्नों में से कम से कम (3) हल करने अनिवार्य हैं। चयन कितने होंगे?
From (15) questions, (10) are to be attempted. At least (3) of the first (5) questions must be attempted. How many selections are possible?
#exam
#question-selection
#atleast
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (875)
B (900)
C (925)
D (950)
Explanation opens after your attempt
Step 1
Concept
Choose (3,4,5) from the first (5) and complete from the remaining (10). \(^{5}C_{3}{}^{10}C_{7}+^{5}C_{4}{}^{10}C_{6}+^{5}C_{5}{}^{10}C_{5}=875\).
Step 2
Why this answer is correct
The correct answer is A. (875). Choose (3,4,5) from the first (5) and complete from the remaining (10). \(^{5}C_{3}{}^{10}C_{7}+^{5}C_{4}{}^{10}C_{6}+^{5}C_{5}{}^{10}C_{5}=875\).
Step 3
Exam Tip
पहले (5) में से (3,4,5) चुनकर बाकी (10) से पूर्ति करें। \(^{5}C_{3}{}^{10}C_{7}+^{5}C_{4}{}^{10}C_{6}+^{5}C_{5}{}^{10}C_{5}=875\)।
Login to save your score, XP, coins and progress. Login
एक परीक्षा में खंड (A) में (6) और खंड (B) में (7) प्रश्न हैं। कुल (8) प्रश्न चुनने हैं और प्रत्येक खंड से कम से कम (3) प्रश्न होने चाहिए। कितने चयन संभव हैं?
An exam has (6) questions in section (A) and (7) in section (B). A total of (8) questions must be chosen with at least (3) from each section. How many selections are possible?
#exam
#sections
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (840)
B (875)
C (910)
D (945)
Explanation opens after your attempt
Step 1
Concept
Cases are (3,5), (4,4), and (5,3). Sum \(^{6}C_{3}{}^{7}C_{5}+^{6}C_{4}{}^{7}C_{4}+^{6}C_{5}{}^{7}C_{3}=945\).
Step 2
Why this answer is correct
The correct answer is D. (945). Cases are (3,5), (4,4), and (5,3). Sum \(^{6}C_{3}{}^{7}C_{5}+^{6}C_{4}{}^{7}C_{4}+^{6}C_{5}{}^{7}C_{3}=945\).
Step 3
Exam Tip
मामले (3,5), (4,4), और (5,3) हैं। योग \(^{6}C_{3}{}^{7}C_{5}+^{6}C_{4}{}^{7}C_{4}+^{6}C_{5}{}^{7}C_{3}=945\)।
Login to save your score, XP, coins and progress. Login
(8) अभ्यर्थियों में से (3) का चयन करना है, पर एक विशेष अभ्यर्थी अवश्य चुना जाए। कितने तरीके होंगे?
From (8) candidates, (3) are to be selected, but one particular candidate must be included. How many ways are possible?
#must-include
#selection
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (15)
B (21)
C (28)
D (35)
Explanation opens after your attempt
Step 1
Concept
Fix the particular candidate and choose remaining (2) from (7). The number is \(^{7}C_{2}=21\).
Step 2
Why this answer is correct
The correct answer is B. (21). Fix the particular candidate and choose remaining (2) from (7). The number is \(^{7}C_{2}=21\).
Step 3
Exam Tip
विशेष अभ्यर्थी को निश्चित मानें और बाकी (2) को (7) में से चुनें। संख्या \(^{7}C_{2}=21\)।
Login to save your score, XP, coins and progress. Login
(9) वैज्ञानिकों में से (4) का दल बनाना है, पर दो विरोधी वैज्ञानिक साथ नहीं हो सकते। कितने दल बनेंगे?
A team of (4) is to be formed from (9) scientists, but two rival scientists cannot be together. How many teams are possible?
#restriction
#team
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (105)
B (110)
C (115)
D (120)
Explanation opens after your attempt
Step 1
Concept
Total selections are \(^{9}C_{4}=126\). Remove \(^{7}C_{2}=21\) selections containing both rivals, answer (105).
Step 2
Why this answer is correct
The correct answer is A. (105). Total selections are \(^{9}C_{4}=126\). Remove \(^{7}C_{2}=21\) selections containing both rivals, answer (105).
Step 3
Exam Tip
कुल \(^{9}C_{4}=126\) हैं। दोनों विरोधी साथ होने पर \(^{7}C_{2}=21\) हटाएं, उत्तर (105)।
Login to save your score, XP, coins and progress. Login
(5) गणित और (4) भौतिकी पुस्तकों में से (5) पुस्तकें चुननी हैं, जिनमें ठीक (2) भौतिकी पुस्तकें हों। कितने चयन हैं?
From (5) mathematics and (4) physics books, (5) books are to be selected with exactly (2) physics books. How many selections are there?
#books
#exactly
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (50)
B (60)
C (70)
D (80)
Explanation opens after your attempt
Step 1
Concept
For exactly (2) physics books use \(^{4}C_{2}\), and for (3) mathematics books use \(^{5}C_{3}\). Product is (60).
Step 2
Why this answer is correct
The correct answer is B. (60). For exactly (2) physics books use \(^{4}C_{2}\), and for (3) mathematics books use \(^{5}C_{3}\). Product is (60).
Step 3
Exam Tip
ठीक (2) भौतिकी के लिए \(^{4}C_{2}\) और (3) गणित के लिए \(^{5}C_{3}\)। गुणन से (60) मिलता है।
Login to save your score, XP, coins and progress. Login
(6) लाल, (5) नीली और (4) हरी गेंदों में से (5) गेंदें चुननी हैं, जिनमें हर रंग की कम से कम एक गेंद हो। कितने चयन संभव हैं?
From (6) red, (5) blue, and (4) green balls, (5) balls are to be selected with at least one ball of each color. How many selections are possible?
#colors
#atleast
#casework
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (1090)
B (1120)
C (1150)
D (1180)
Explanation opens after your attempt
Step 1
Concept
Use all ordered color distributions of (3,1,1) and (2,2,1). The total is (1150).
Step 2
Why this answer is correct
The correct answer is C. (1150). Use all ordered color distributions of (3,1,1) and (2,2,1). The total is (1150).
Step 3
Exam Tip
रंग-वितरण (3,1,1) और (2,2,1) के सभी क्रम लें। योग (1150) आता है।
Login to save your score, XP, coins and progress. Login
(18) सदस्यों में से (6) की समिति बनानी है। अध्यक्ष और सचिव समिति के सामान्य सदस्य के रूप में साथ नहीं रह सकते। कितनी समितियां बनेंगी?
From (18) members, a committee of (6) is to be formed. The president and secretary cannot both be in the committee as ordinary members. How many committees are possible?
#committee
#error-check
#exclusion
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (16082)
B (16128)
C (16250)
D (16302)
Explanation opens after your attempt
Correct Answer
B. (16128)
Step 1
Concept
Total is \(^{18}C_{6}=18564\), and selections with both special members are \(^{16}C_{4}=1820\), giving (16744). None of the options is correct.
Step 2
Why this answer is correct
The correct answer is B. (16128). Total is \(^{18}C_{6}=18564\), and selections with both special members are \(^{16}C_{4}=1820\), giving (16744). None of the options is correct.
Step 3
Exam Tip
कुल \(^{18}C_{6}=18564\) से दोनों विशेष शामिल वाले \(^{16}C_{4}=1820\) नहीं, सही हटाना यही है और उत्तर (16744) होगा। दिए विकल्पों में सही उत्तर नहीं है।
Login to save your score, XP, coins and progress. Login
(20) विद्यार्थियों में से (6) चुनने हैं। दो विशेष विद्यार्थी में से ठीक एक चुना जाना चाहिए। कितने चयन होंगे?
From (20) students, (6) are to be chosen. Exactly one of two special students must be selected. How many selections are possible?
#exactly-one
#selection
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (17136)
B (17424)
C (18144)
D (18564)
Explanation opens after your attempt
Correct Answer
C. (18144)
Step 1
Concept
Choose (1) of the two special students and (5) from the remaining (18). The number is \(^{2}C_{1}{}^{18}C_{5}=17136\).
Step 2
Why this answer is correct
The correct answer is C. (18144). Choose (1) of the two special students and (5) from the remaining (18). The number is \(^{2}C_{1}{}^{18}C_{5}=17136\).
Step 3
Exam Tip
दो विशेष में से (1) चुनें और बाकी (5) को (18) में से चुनें। संख्या \(^{2}C_{1}{}^{18}C_{5}=17136\) होती है।
Login to save your score, XP, coins and progress. Login
यदि (^{2n}C_{3}=2,^{n}C_{3}+n^{2}(n-1)) है, तो कथन किसके लिए सही है?
If (^{2n}C_{3}=2,^{n}C_{3}+n^{2}(n-1)), for which condition is the statement true?
#identity
#combinatorial-proof
#ncr
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A सभी \(n\geq 3\) / all \(n\geq 3\)
B केवल (n=3) / only (n=3)
C केवल सम (n) / only even (n)
D कभी नहीं / never
Explanation opens after your attempt
Correct Answer
A. सभी \(n\geq 3\) / all \(n\geq 3\)
Step 1
Concept
While choosing (3) from two groups of size (n), either all (3) come from one group or the split is (2,1). This proves the identity.
Step 2
Why this answer is correct
The correct answer is A. सभी \(n\geq 3\) / all \(n\geq 3\). While choosing (3) from two groups of size (n), either all (3) come from one group or the split is (2,1). This proves the identity.
Step 3
Exam Tip
दो (n) आकार के समूहों से (3) चुनने में सभी (3) एक समूह से या (2,1) विभाजन से आते हैं। यही पहचान है।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{r}=^{n}C_{r-2}\) और (r>2) है, तो (n) किसके बराबर है?
If \(^{n}C_{r}=^{n}C_{r-2}\) and (r>2), then (n) is equal to what?
#ncr
#symmetry
#formula
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (2r-2)
B (2r-1)
C (2r)
D (2r+1)
Explanation opens after your attempt
Step 1
Concept
For same (n), the complementary indices add to (n). Hence (r+(r-2)=n).
Step 2
Why this answer is correct
The correct answer is A. (2r-2). For same (n), the complementary indices add to (n). Hence (r+(r-2)=n).
Step 3
Exam Tip
समान (n) में सूचकांकों का योग (n) होता है। इसलिए (r+(r-2)=n)।
Login to save your score, XP, coins and progress. Login
(14) बिंदुओं में से (6) बिंदु एक रेखा पर और (4) दूसरे रेखा पर हैं। दोनों रेखाएं अलग हैं और अन्य कोई (3) कोलिनियर नहीं। कितने त्रिभुज बनेंगे?
Among (14) points, (6) lie on one line and (4) on another line. The lines are distinct and no other (3) points are collinear. How many triangles can be formed?
#geometry
#collinear
#triangles
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (320)
B (324)
C (328)
D (332)
Explanation opens after your attempt
Step 1
Concept
Total is \(^{14}C_{3}=364\). Invalid selections are \(^{6}C_{3}+^{4}C_{3}=24\), so the answer is (340).
Step 2
Why this answer is correct
The correct answer is B. (324). Total is \(^{14}C_{3}=364\). Invalid selections are \(^{6}C_{3}+^{4}C_{3}=24\), so the answer is (340).
Step 3
Exam Tip
कुल \(^{14}C_{3}=364\) है। अमान्य चयन \(^{6}C_{3}+^{4}C_{3}=24\), इसलिए उत्तर (340) नहीं बल्कि (340) है।
Login to save your score, XP, coins and progress. Login
(10) अलग-अलग बिंदुओं में से (4) बिंदु एक रेखा पर हैं। कितनी सीधी रेखाएं निर्धारित होंगी?
Among (10) distinct points, (4) points are collinear. How many straight lines are determined?
#geometry
#lines
#collinear
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (37)
B (38)
C (39)
D (40)
Explanation opens after your attempt
Step 1
Concept
Total pairs are \(^{10}C_{2}=45\). The \(^{4}C_{2}=6\) pairs among collinear points give only (1) line, so (45-6+1=40).
Step 2
Why this answer is correct
The correct answer is D. (40). Total pairs are \(^{10}C_{2}=45\). The \(^{4}C_{2}=6\) pairs among collinear points give only (1) line, so (45-6+1=40).
Step 3
Exam Tip
कुल युग्म \(^{10}C_{2}=45\) हैं। (4) कोलिनियर बिंदुओं के \(^{4}C_{2}=6\) युग्मों की जगह (1) रेखा होगी, इसलिए (45-6+1=40)।
Login to save your score, XP, coins and progress. Login
एक \(5\times 4\) ग्रिड में बाएं-नीचे से दाएं-ऊपर जाने के लिए केवल दाएं और ऊपर चालें चलनी हैं। कुल कितने रास्ते हैं?
In a \(5\times 4\) grid, one moves from bottom-left to top-right using only right and up moves. How many paths are possible?
#grid
#path
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (126)
B (120)
C (100)
D (84)
Explanation opens after your attempt
Step 1
Concept
There are (9) moves with (5) right and (4) up. Paths are \(^{9}C_{5}=126\).
Step 2
Why this answer is correct
The correct answer is A. (126). There are (9) moves with (5) right and (4) up. Paths are \(^{9}C_{5}=126\).
Step 3
Exam Tip
कुल चालें (9) हैं जिनमें (5) दाएं और (4) ऊपर हैं। रास्ते \(^{9}C_{5}=126\) हैं।
Login to save your score, XP, coins and progress. Login
\(6\times 5\) ग्रिड में रास्ते में एक निर्धारित बिंदु तक जरूर जाना है, जिसके लिए पहले (2) दाएं और (3) ऊपर चालें चाहिए। कुल रास्ते कितने हैं?
In a \(6\times 5\) grid, a path must pass through a fixed point that requires (2) right and (3) up moves first. How many paths are possible?
#grid
#path
#through-point
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (560)
B (700)
C (840)
D (1050)
Explanation opens after your attempt
Step 1
Concept
First part has \(^{5}C_{2}=10\) paths and the second part has \(^{6}C_{4}=15\) paths. Total is (150).
Step 2
Why this answer is correct
The correct answer is B. (700). First part has \(^{5}C_{2}=10\) paths and the second part has \(^{6}C_{4}=15\) paths. Total is (150).
Step 3
Exam Tip
पहले भाग के रास्ते \(^{5}C_{2}=10\) और दूसरे भाग के रास्ते \(^{6}C_{4}=15\) नहीं, सही दूसरा \(^{6}C_{4}=15\) है। कुल (150) होता है।
Login to save your score, XP, coins and progress. Login
(1) से (20) तक की संख्याओं में से (4) संख्याएं चुननी हैं ताकि कोई दो लगातार न हों। कितने चयन संभव हैं?
Choose (4) numbers from (1) to (20) so that no two are consecutive. How many selections are possible?
#nonconsecutive
#selection
#formula
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (2380)
B (3060)
C (3876)
D (4845)
Explanation opens after your attempt
Step 1
Concept
If no two are consecutive, use \(^{n-r+1}C_{r}\). Here \(^{17}C_{4}=2380\), so the correct answer is (2380).
Step 2
Why this answer is correct
The correct answer is C. (3876). If no two are consecutive, use \(^{n-r+1}C_{r}\). Here \(^{17}C_{4}=2380\), so the correct answer is (2380).
Step 3
Exam Tip
कोई दो लगातार न हों तो सूत्र \(^{n-r+1}C_{r}\) है। यहां \(^{17}C_{4}=2380\), इसलिए सही विकल्प (2380) है।
Login to save your score, XP, coins and progress. Login
(1) से (15) तक की संख्याओं में से (5) संख्याएं चुननी हैं, जिनमें कोई दो लगातार न हों। कितने तरीके हैं?
Choose (5) numbers from (1) to (15) such that no two are consecutive. How many ways are possible?
#nonconsecutive
#gap-method
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (462)
B (495)
C (540)
D (600)
Explanation opens after your attempt
Step 1
Concept
By the gap method, the count is \(^{15-5+1}C_{5}=^{11}C_{5}=462\). In such problems identify the gap condition first.
Step 2
Why this answer is correct
The correct answer is A. (462). By the gap method, the count is \(^{15-5+1}C_{5}=^{11}C_{5}=462\). In such problems identify the gap condition first.
Step 3
Exam Tip
अलगाव विधि से संख्या \(^{15-5+1}C_{5}=^{11}C_{5}=462\)। ऐसी समस्याओं में पहले अंतर की शर्त पहचानें।
Login to save your score, XP, coins and progress. Login
(12) लोगों में से (5) चुनने हैं, पर तीन विशेष लोगों में से कम से कम (2) चुने जाएं। कितने चयन होंगे?
From (12) people, (5) are to be chosen, but at least (2) of (3) special people must be selected. How many selections are possible?
#atleast
#special
#selection
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (225)
B (250)
C (276)
D (300)
Explanation opens after your attempt
Step 1
Concept
Cases are (2) special and (3) others, or (3) special and (2) others. The value is \(^{3}C_{2}{}^{9}C_{3}+^{3}C_{3}{}^{9}C_{2}=288\).
Step 2
Why this answer is correct
The correct answer is A. (225). Cases are (2) special and (3) others, or (3) special and (2) others. The value is \(^{3}C_{2}{}^{9}C_{3}+^{3}C_{3}{}^{9}C_{2}=288\).
Step 3
Exam Tip
मामले (2) विशेष और (3) अन्य, या (3) विशेष और (2) अन्य हैं। \(^{3}C_{2}{}^{9}C_{3}+^{3}C_{3}{}^{9}C_{2}=288\) होता है।
Login to save your score, XP, coins and progress. Login
(9) प्रश्नों में से (5) चुनने हैं, लेकिन प्रश्न (1) और प्रश्न (2) में से कम से कम एक चुना जाना चाहिए। कितने चयन संभव हैं?
From (9) questions, (5) are to be chosen, but at least one of question (1) and question (2) must be chosen. How many selections are possible?
#atleast-one
#exam
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (91)
B (98)
C (105)
D (112)
Explanation opens after your attempt
Step 1
Concept
Subtract cases where neither is chosen, \(^{7}C_{5}=21\), from total \(^{9}C_{5}=126\). The answer is (105).
Step 2
Why this answer is correct
The correct answer is A. (91). Subtract cases where neither is chosen, \(^{7}C_{5}=21\), from total \(^{9}C_{5}=126\). The answer is (105).
Step 3
Exam Tip
कुल \(^{9}C_{5}=126\) से दोनों न चुने जाने वाले \(^{7}C_{5}=21\) घटते हैं। उत्तर (105) है।
Login to save your score, XP, coins and progress. Login
(7) शिक्षकों और (8) छात्रों में से (6) सदस्यीय दल बनाना है, जिसमें शिक्षकों की संख्या छात्रों से अधिक हो। कितने दल बनेंगे?
From (7) teachers and (8) students, a (6)-member team is to be formed in which teachers are more than students. How many teams are possible?
#team
#comparison
#casework
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (917)
B (980)
C (1050)
D (1120)
Explanation opens after your attempt
Step 1
Concept
Cases are (4T2S), (5T1S), and (6T0S). Sum is \(^{7}C_{4}{}^{8}C_{2}+^{7}C_{5}{}^{8}C_{1}+^{7}C_{6}=1155\).
Step 2
Why this answer is correct
The correct answer is A. (917). Cases are (4T2S), (5T1S), and (6T0S). Sum is \(^{7}C_{4}{}^{8}C_{2}+^{7}C_{5}{}^{8}C_{1}+^{7}C_{6}=1155\).
Step 3
Exam Tip
शिक्षक अधिक होने के मामले (4T2S), (5T1S), (6T0S) हैं। योग \(^{7}C_{4}{}^{8}C_{2}+^{7}C_{5}{}^{8}C_{1}+^{7}C_{6}=1155\) होता है।
Login to save your score, XP, coins and progress. Login
(16) वस्तुओं में (10) अलग और (6) एक जैसी हैं। (4) वस्तुएं चुनने के कितने तरीके हैं?
Among (16) objects, (10) are distinct and (6) are identical. How many ways are there to select (4) objects?
#identical-objects
#selection
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (210)
B (286)
C (375)
D (385)
Explanation opens after your attempt
Step 1
Concept
We can take (0) to (4) identical objects. The total is \(^{10}C_{4}+^{10}C_{3}+^{10}C_{2}+^{10}C_{1}+1=386\).
Step 2
Why this answer is correct
The correct answer is D. (385). We can take (0) to (4) identical objects. The total is \(^{10}C_{4}+^{10}C_{3}+^{10}C_{2}+^{10}C_{1}+1=386\).
Step 3
Exam Tip
एक जैसी वस्तुओं में से (0) से (4) तक ली जा सकती हैं। योग \(\sum_{k=0}^{4}{}^{10}C_{4-k}=386\) नहीं, सही \(^{10}C_{4}+^{10}C_{3}+^{10}C_{2}+^{10}C_{1}+1=386\) है।
Login to save your score, XP, coins and progress. Login
(7) अलग-अलग फलों और (5) एक जैसी टॉफियों में से कुल (4) वस्तुएं चुननी हैं। कितने चयन संभव हैं?
From (7) distinct fruits and (5) identical candies, (4) objects are to be selected. How many selections are possible?
#identical
#distinct
#selection
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (64)
B (99)
C (120)
D (140)
Explanation opens after your attempt
Step 1
Concept
The number of identical candies can be (0) to (4). Sum \(^{7}C_{4}+^{7}C_{3}+^{7}C_{2}+^{7}C_{1}+1=99\).
Step 2
Why this answer is correct
The correct answer is B. (99). The number of identical candies can be (0) to (4). Sum \(^{7}C_{4}+^{7}C_{3}+^{7}C_{2}+^{7}C_{1}+1=99\).
Step 3
Exam Tip
एक जैसी टॉफियों की संख्या (0) से (4) तक हो सकती है। योग \(^{7}C_{4}+^{7}C_{3}+^{7}C_{2}+^{7}C_{1}+1=99\)।
Login to save your score, XP, coins and progress. Login
कितने तरीकों से (5) पुरस्कार (12) विद्यार्थियों में बांटे जा सकते हैं यदि किसी विद्यार्थी को अधिकतम एक पुरस्कार मिले और पुरस्कार समान हों?
In how many ways can (5) identical prizes be distributed among (12) students if each student gets at most one prize?
#distribution
#identical-prizes
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (792)
B (950)
C (1584)
D (2484)
Explanation opens after your attempt
Step 1
Concept
Identical prizes with at most one per student means choosing (5) students. The number is \(^{12}C_{5}=792\).
Step 2
Why this answer is correct
The correct answer is A. (792). Identical prizes with at most one per student means choosing (5) students. The number is \(^{12}C_{5}=792\).
Step 3
Exam Tip
समान पुरस्कार और अधिकतम एक पुरस्कार का अर्थ है केवल (5) विद्यार्थियों का चयन। संख्या \(^{12}C_{5}=792\) है।
Login to save your score, XP, coins and progress. Login
(8) लोगों की सभा में कितने हाथ मिलाने होंगे यदि हर जोड़ा ठीक एक बार हाथ मिलाए?
In a meeting of (8) people, how many handshakes occur if every pair shakes hands exactly once?
#handshake
#pairs
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (24)
B (28)
C (32)
D (36)
Explanation opens after your attempt
Step 1
Concept
Each handshake corresponds to choosing (2) people. Hence the number is \(^{8}C_{2}=28\).
Step 2
Why this answer is correct
The correct answer is B. (28). Each handshake corresponds to choosing (2) people. Hence the number is \(^{8}C_{2}=28\).
Step 3
Exam Tip
हर हाथ मिलाना (2) लोगों के चयन के बराबर है। इसलिए संख्या \(^{8}C_{2}=28\)।
Login to save your score, XP, coins and progress. Login
(30) लोगों की सभा में (435) हाथ मिलाए गए। यदि हर जोड़ा एक बार हाथ मिलाता है, तो कितने लोग अनुपस्थित थे जब कुल आमंत्रित (35) थे?
In a gathering, (435) handshakes occurred. If every pair shook hands once and (35) people were invited, how many were absent?
#handshake
#equation
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
From \(^{n}C_{2}=435\), (n=30) people were present. Therefore absent people are (35-30=5).
Step 2
Why this answer is correct
The correct answer is C. (5). From \(^{n}C_{2}=435\), (n=30) people were present. Therefore absent people are (35-30=5).
Step 3
Exam Tip
\(^{n}C_{2}=435\) से (n=30) उपस्थित हैं। अतः अनुपस्थित (35-30=5)।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{0}+^{n}C_{1}+^{n}C_{2}=46\) है, तो (n) का मान क्या है?
If \(^{n}C_{0}+^{n}C_{1}+^{n}C_{2}=46\), what is the value of (n)?
#ncr-sum
#equation
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The equation is (1+n+\frac{n(n-1)}{2}=46). Solving gives (n=9).
Step 2
Why this answer is correct
The correct answer is B. (9). The equation is (1+n+\frac{n(n-1)}{2}=46). Solving gives (n=9).
Step 3
Exam Tip
समीकरण (1+n+\frac{n(n-1)}{2}=46) है। इससे (n=9) मिलता है।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{2}+^{n}C_{3}=^{n+1}C_{3}\) है, तो यह पहचान किस नियम पर आधारित है?
If \(^{n}C_{2}+^{n}C_{3}=^{n+1}C_{3}\), this identity is based on which rule?
#pascal-rule
#identity
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A पूरक नियम / Complement rule
B पास्कल नियम / Pascal rule
C गुणन नियम / Multiplication rule
D समावेशन-बहिष्करण / Inclusion-exclusion
Explanation opens after your attempt
Correct Answer
B. पास्कल नियम / Pascal rule
Step 1
Concept
This is the form \(^{n}C_{r-1}+^{n}C_{r}=^{n+1}C_{r}\). It is called Pascal's rule.
Step 2
Why this answer is correct
The correct answer is B. पास्कल नियम / Pascal rule. This is the form \(^{n}C_{r-1}+^{n}C_{r}=^{n+1}C_{r}\). It is called Pascal's rule.
Step 3
Exam Tip
यह \(^{n}C_{r-1}+^{n}C_{r}=^{n+1}C_{r}\) का रूप है। इसे पास्कल नियम कहते हैं।
Login to save your score, XP, coins and progress. Login
\(^{15}C_{7}+^{15}C_{8}\) का सरल मान किसके बराबर है?
The simplified value of \(^{15}C_{7}+^{15}C_{8}\) is equal to what?
#symmetry
#pascal
#ncr
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A \(^{16}C_{7}\)
B \(^{16}C_{8}\)
C \(^{30}C_{15}\)
D \(2,^{15}C_{7}\)
Explanation opens after your attempt
Correct Answer
D. \(2,^{15}C_{7}\)
Step 1
Concept
Since \(^{15}C_{8}=^{15}C_{7}\), the sum is \(2,^{15}C_{7}\). By Pascal's rule it is also \(^{16}C_{8}\).
Step 2
Why this answer is correct
The correct answer is D. \(2,^{15}C_{7}\). Since \(^{15}C_{8}=^{15}C_{7}\), the sum is \(2,^{15}C_{7}\). By Pascal's rule it is also \(^{16}C_{8}\).
Step 3
Exam Tip
क्योंकि \(^{15}C_{8}=^{15}C_{7}\), योग \(2,^{15}C_{7}\) है। पास्कल से यह \(^{16}C_{8}\) भी है।
Login to save your score, XP, coins and progress. Login
(10) अलग-अलग वस्तुओं में से (3) या (4) वस्तुएं चुनने के कुल तरीके कितने हैं?
How many total ways are there to choose (3) or (4) objects from (10) distinct objects?
#or-rule
#selection
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (330)
B (340)
C (350)
D (360)
Explanation opens after your attempt
Step 1
Concept
The selection count is \(^{10}C_{3}+^{10}C_{4}=120+210=330\). Here OR means addition.
Step 2
Why this answer is correct
The correct answer is A. (330). The selection count is \(^{10}C_{3}+^{10}C_{4}=120+210=330\). Here OR means addition.
Step 3
Exam Tip
चयन \(^{10}C_{3}+^{10}C_{4}=120+210=330\) है। या का अर्थ यहां जोड़ना है।
Login to save your score, XP, coins and progress. Login
(13) छात्रों में से (5) चुनने हैं, पर एक निश्चित छात्र न चुना जाए। कितने तरीके हैं?
From (13) students, (5) are to be selected, but one fixed student must not be selected. How many ways are possible?
#not-include
#selection
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (792)
B (924)
C (1287)
D (1716)
Explanation opens after your attempt
Step 1
Concept
Remove the forbidden student and choose (5) from (12). The number is \(^{12}C_{5}=792\).
Step 2
Why this answer is correct
The correct answer is A. (792). Remove the forbidden student and choose (5) from (12). The number is \(^{12}C_{5}=792\).
Step 3
Exam Tip
न चुने जाने वाले छात्र को हटाकर (12) में से (5) चुनें। संख्या \(^{12}C_{5}=792\)।
Login to save your score, XP, coins and progress. Login
एक बैग में (5) अलग लाल और (7) अलग काली गेंदें हैं। (4) गेंदें चुननी हैं जिनमें रंग समान न हो। कितने चयन होंगे?
A bag has (5) distinct red and (7) distinct black balls. (4) balls are to be selected such that not all are of the same color. How many selections are possible?
#balls
#color
#exclusion
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (410)
B (430)
C (455)
D (475)
Explanation opens after your attempt
Step 1
Concept
Subtract all-red \(^{5}C_{4}=5\) and all-black \(^{7}C_{4}=35\) from total \(^{12}C_{4}=495\). Answer (455).
Step 2
Why this answer is correct
The correct answer is C. (455). Subtract all-red \(^{5}C_{4}=5\) and all-black \(^{7}C_{4}=35\) from total \(^{12}C_{4}=495\). Answer (455).
Step 3
Exam Tip
कुल \(^{12}C_{4}=495\) से सभी लाल \(^{5}C_{4}=5\) और सभी काली \(^{7}C_{4}=35\) हटाएं। उत्तर (455)।
Login to save your score, XP, coins and progress. Login
(9) अंकों \(1,2,\ldots,9\) में से (4) अंक चुनने हैं जिनका योग सम हो। कितने चयन हैं?
Choose (4) digits from \(1,2,\ldots,9\) such that their sum is even. How many selections are there?
#parity
#digits
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (56)
B (60)
C (64)
D (70)
Explanation opens after your attempt
Step 1
Concept
For an even sum, the number of odd digits must be (0,2,4). Count \(^{4}C_{4}+^{5}C_{2}{}^{4}C_{2}+^{5}C_{4}=66\).
Step 2
Why this answer is correct
The correct answer is C. (64). For an even sum, the number of odd digits must be (0,2,4). Count \(^{4}C_{4}+^{5}C_{2}{}^{4}C_{2}+^{5}C_{4}=66\).
Step 3
Exam Tip
सम योग के लिए विषम अंकों की संख्या (0,2,4) होनी चाहिए। गणना \(^{4}C_{4}+^{5}C_{2}{}^{4}C_{2}+^{5}C_{4}=66\) है।
Login to save your score, XP, coins and progress. Login
(1) से (12) तक की संख्याओं में से (3) संख्याएं चुननी हैं जिनका योग विषम हो। कितने चयन संभव हैं?
Choose (3) numbers from (1) to (12) such that their sum is odd. How many selections are possible?
#parity
#numbers
#casework
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (100)
B (110)
C (120)
D (130)
Explanation opens after your attempt
Step 1
Concept
There are (6) odd and (6) even numbers. For odd sum choose (1) or (3) odd numbers, giving \(^{6}C_{1}{}^{6}C_{2}+^{6}C_{3}=110\).
Step 2
Why this answer is correct
The correct answer is B. (110). There are (6) odd and (6) even numbers. For odd sum choose (1) or (3) odd numbers, giving \(^{6}C_{1}{}^{6}C_{2}+^{6}C_{3}=110\).
Step 3
Exam Tip
(6) विषम और (6) सम संख्याएं हैं। विषम योग के लिए (1) या (3) विषम चुनें, संख्या \(^{6}C_{1}{}^{6}C_{2}+^{6}C_{3}=110\)।
Login to save your score, XP, coins and progress. Login
(6) अलग-अलग रेखाएं और (5) अलग-अलग वृत्त हैं। दो आकृतियां चुननी हैं जिनमें कम से कम एक रेखा हो। कितने चयन हैं?
There are (6) distinct lines and (5) distinct circles. Two figures are to be selected with at least one line. How many selections are possible?
#figures
#atleast-one
#combinations
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (45)
B (50)
C (55)
D (60)
Explanation opens after your attempt
Step 1
Concept
Subtract circle-only selections \(^{5}C_{2}=10\) from total \(^{11}C_{2}=55\). The correct count is (45).
Step 2
Why this answer is correct
The correct answer is C. (55). Subtract circle-only selections \(^{5}C_{2}=10\) from total \(^{11}C_{2}=55\). The correct count is (45).
Step 3
Exam Tip
कुल \(^{11}C_{2}=55\) से केवल वृत्त \(^{5}C_{2}=10\) घटाने चाहिए। सही संख्या (45) है।
Login to save your score, XP, coins and progress. Login
(11) खिलाड़ियों में से (7) चुनने हैं, और (4) विशेष खिलाड़ियों में से अधिकतम (2) चुने जा सकते हैं। कितने चयन होंगे?
From (11) players, (7) are to be selected, and at most (2) of (4) special players can be selected. How many selections are possible?
#atmost
#special
#selection
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (126)
B (147)
C (168)
D (189)
Explanation opens after your attempt
Step 1
Concept
The number of special players can be (0,1,2), with (7) ordinary players available. Sum \(^{4}C_{0}{}^{7}C_{7}+^{4}C_{1}{}^{7}C_{6}+^{4}C_{2}{}^{7}C_{5}=155\).
Step 2
Why this answer is correct
The correct answer is A. (126). The number of special players can be (0,1,2), with (7) ordinary players available. Sum \(^{4}C_{0}{}^{7}C_{7}+^{4}C_{1}{}^{7}C_{6}+^{4}C_{2}{}^{7}C_{5}=155\).
Step 3
Exam Tip
विशेष खिलाड़ियों की संख्या (0,1,2) हो सकती है, पर (7) अन्य ही हैं। योग \(^{4}C_{0}{}^{7}C_{7}+^{4}C_{1}{}^{7}C_{6}+^{4}C_{2}{}^{7}C_{5}=155\) है।
Login to save your score, XP, coins and progress. Login
यदि \(^{n}C_{r}=^{n}C_{r+1}\) है, तो (n) और (r) के बीच संबंध क्या है?
If \(^{n}C_{r}=^{n}C_{r+1}\), what is the relation between (n) and (r)?
#ncr
#equality
#relation
50 50-50 2 wrong hide
⏭ Skip Next question
+10 Time+ 10 sec extra
? Hint Small clue
A (n=2r)
B (n=2r+1)
C (n=2r+2)
D (n=r+1)
Explanation opens after your attempt
Correct Answer
B. (n=2r+1)
Step 1
Concept
For equality, complementary indices add to (n). Thus (r+(r+1)=n), so (n=2r+1).
Step 2
Why this answer is correct
The correct answer is B. (n=2r+1). For equality, complementary indices add to (n). Thus (r+(r+1)=n), so (n=2r+1).
Step 3
Exam Tip
समानता में पूरक सूचकांक का योग (n) है। इसलिए (r+(r+1)=n), अतः (n=2r+1)।
Login to save your score, XP, coins and progress. Login
