(10) बिंदुओं में से कोई (3) एक सीध में नहीं हैं। उनसे कितने वृत्त निर्धारित होंगे?
There are (10) points, no (3) of which are collinear. How many circles will be determined?
Explanation opens after your attempt
Step 1
Concept
A circle is determined by (3) points. Thus \(\binom{10}{3}=120\).
Step 2
Why this answer is correct
The correct answer is C. (120). A circle is determined by (3) points. Thus \(\binom{10}{3}=120\).
Step 3
Exam Tip
एक वृत्त (3) बिंदुओं से निर्धारित होता है। अतः \(\binom{10}{3}=120\) होगा।
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(12) बिंदुओं से कितनी सीधी रेखाएं बनेंगी यदि कोई (3) बिंदु एक ही रेखा पर नहीं हैं?
How many straight lines can be formed from (12) points if no (3) points lie on the same line?
Explanation opens after your attempt
Step 1
Concept
A line is formed by (2) points. Hence \(\binom{12}{2}=66\) is correct.
Step 2
Why this answer is correct
The correct answer is B. (66). A line is formed by (2) points. Hence \(\binom{12}{2}=66\) is correct.
Step 3
Exam Tip
एक रेखा (2) बिंदुओं से बनती है। इसलिए \(\binom{12}{2}=66\) सही है।
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(14) बिंदुओं में से (3) बिंदु चुनकर कितने त्रिभुज बन सकते हैं यदि कोई (3) बिंदु एक सीध में नहीं हैं?
How many triangles can be formed by choosing (3) points from (14) points if no (3) points are collinear?
Explanation opens after your attempt
Step 1
Concept
A triangle needs (3) points. Hence \(\binom{14}{3}=364\).
Step 2
Why this answer is correct
The correct answer is C. (364). A triangle needs (3) points. Hence \(\binom{14}{3}=364\).
Step 3
Exam Tip
त्रिभुज के लिए (3) बिंदु चाहिए। इसलिए \(\binom{14}{3}=364\) है।
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(11) टीमों की लीग में हर टीम दूसरी टीम से एक बार खेले तो कुल मैच कितने होंगे?
In a league of (11) teams, if each team plays every other team once, how many matches will be played?
Explanation opens after your attempt
Step 1
Concept
Each match is formed by a pair of (2) teams. The total matches are \(\binom{11}{2}=55\).
Step 2
Why this answer is correct
The correct answer is B. (55). Each match is formed by a pair of (2) teams. The total matches are \(\binom{11}{2}=55\).
Step 3
Exam Tip
हर मैच (2) टीमों की जोड़ी से बनता है। कुल मैच \(\binom{11}{2}=55\) होंगे।
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(20) व्यक्तियों में से (2) व्यक्तियों के बीच हाथ मिलाने की अधिकतम संख्या क्या है?
What is the maximum number of handshakes among (20) persons?
Explanation opens after your attempt
Step 1
Concept
Each handshake is a pair of (2) persons. Therefore \(\binom{20}{2}=190\).
Step 2
Why this answer is correct
The correct answer is A. (190). Each handshake is a pair of (2) persons. Therefore \(\binom{20}{2}=190\).
Step 3
Exam Tip
हर हाथ मिलाना (2) व्यक्तियों की एक जोड़ी है। इसलिए \(\binom{20}{2}=190\) होगा।
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(5) हिंदी और (6) संस्कृत पुस्तकों में से (2) हिंदी और (3) संस्कृत पुस्तकें कितने तरीकों से चुनी जा सकती हैं?
From (5) Hindi and (6) Sanskrit books, in how many ways can (2) Hindi and (3) Sanskrit books be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{5}{2}\binom{6}{3}=10\cdot20=200\). Multiply selections from different subjects.
Step 2
Why this answer is correct
The correct answer is D. (200). The ways are \(\binom{5}{2}\binom{6}{3}=10\cdot20=200\). Multiply selections from different subjects.
Step 3
Exam Tip
तरीके \(\binom{5}{2}\binom{6}{3}=10\cdot20=200\) हैं। अलग विषयों के चयन को गुणा करें।
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(7) डॉक्टरों और (4) नर्सों में से (2) डॉक्टर और (1) नर्स की टीम कितने तरीकों से बनेगी?
In how many ways can a team of (2) doctors and (1) nurse be formed from (7) doctors and (4) nurses?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{7}{2}\binom{4}{1}=21\cdot4=84\). Count each condition separately.
Step 2
Why this answer is correct
The correct answer is C. (84). The ways are \(\binom{7}{2}\binom{4}{1}=21\cdot4=84\). Count each condition separately.
Step 3
Exam Tip
तरीके \(\binom{7}{2}\binom{4}{1}=21\cdot4=84\) हैं। शर्तों को अलग-अलग गिनें।
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(8) लाल और (7) सफेद फूलों में से (2) लाल और (2) सफेद फूल कितने तरीकों से चुने जा सकते हैं?
From (8) red and (7) white flowers, in how many ways can (2) red and (2) white flowers be selected?
Explanation opens after your attempt
Step 1
Concept
The selection is \(\binom{8}{2}\binom{7}{2}=28\cdot21=588\). Use a separate combination for each color.
Step 2
Why this answer is correct
The correct answer is B. (588). The selection is \(\binom{8}{2}\binom{7}{2}=28\cdot21=588\). Use a separate combination for each color.
Step 3
Exam Tip
चयन \(\binom{8}{2}\binom{7}{2}=28\cdot21=588\) है। हर रंग के लिए अलग संयोजन लगाएं।
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(6) लड़कों और (5) लड़कियों में से (3) लड़के और (1) लड़की कितने तरीकों से चुने जा सकते हैं?
From (6) boys and (5) girls, in how many ways can (3) boys and (1) girl be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{6}{3}\binom{5}{1}=20\cdot5=100\). Multiply selections from different groups.
Step 2
Why this answer is correct
The correct answer is A. (100). The ways are \(\binom{6}{3}\binom{5}{1}=20\cdot5=100\). Multiply selections from different groups.
Step 3
Exam Tip
तरीके \(\binom{6}{3}\binom{5}{1}=20\cdot5=100\) हैं। अलग वर्गों से चयन में गुणा करें।
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(9) अलग-अलग फाइलों में से (6) फाइलें चुनने के कितने तरीके हैं?
How many ways are there to choose (6) files from (9) different files?
Explanation opens after your attempt
Step 1
Concept
\(\binom{9}{6}=\binom{9}{3}=84\). Complementary selection makes calculation easier.
Step 2
Why this answer is correct
The correct answer is D. (84). \(\binom{9}{6}=\binom{9}{3}=84\). Complementary selection makes calculation easier.
Step 3
Exam Tip
\(\binom{9}{6}=\binom{9}{3}=84\) है। पूरक चयन से गणना आसान होती है।
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(13) अलग-अलग अध्यायों में से (3) अध्याय पुनरावृत्ति के लिए कितने तरीकों से चुने जा सकते हैं?
How many ways can (3) chapters be selected from (13) different chapters for revision?
Explanation opens after your attempt
Step 1
Concept
Only selection of chapters is involved. Hence \(\binom{13}{3}=286\) is correct.
Step 2
Why this answer is correct
The correct answer is C. (286). Only selection of chapters is involved. Hence \(\binom{13}{3}=286\) is correct.
Step 3
Exam Tip
अध्यायों का केवल चयन है। इसलिए \(\binom{13}{3}=286\) सही है।
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(10) स्वयंसेवकों में से (5) स्वयंसेवकों की ड्यूटी टीम कितने तरीकों से चुनी जा सकती है?
In how many ways can a duty team of (5) volunteers be selected from (10) volunteers?
Explanation opens after your attempt
Step 1
Concept
Selecting a team is a combination, so \(\binom{10}{5}=252\). If posts are not assigned, do not consider order.
Step 2
Why this answer is correct
The correct answer is B. (252). Selecting a team is a combination, so \(\binom{10}{5}=252\). If posts are not assigned, do not consider order.
Step 3
Exam Tip
टीम चुनना संयोजन है इसलिए \(\binom{10}{5}=252\) होगा। पद न दिए हों तो क्रम नहीं देखें।
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(18) अलग-अलग बैजों में से (2) बैज चुनने के कितने तरीके हैं?
How many ways are there to choose (2) badges from (18) different badges?
Explanation opens after your attempt
Step 1
Concept
The number of ways is \(\binom{18}{2}=153\). Order is not counted in selection.
Step 2
Why this answer is correct
The correct answer is A. (153). The number of ways is \(\binom{18}{2}=153\). Order is not counted in selection.
Step 3
Exam Tip
दो बैज चुनने के तरीके \(\binom{18}{2}=153\) हैं। चयन में क्रम नहीं गिना जाता।
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(5) लड़कों और (3) लड़कियों में से कुल (3) विद्यार्थी चुनने हैं जिनमें कम से कम (1) लड़की हो। कितने तरीके हैं?
From (5) boys and (3) girls, (3) students are to be selected with at least (1) girl. How many ways are there?
Explanation opens after your attempt
Step 1
Concept
Total ways are \(\binom{8}{3}=56\), and all-boy selections are \(\binom{5}{3}=10\). Therefore (56-10=46) ways.
Step 2
Why this answer is correct
The correct answer is C. (46). Total ways are \(\binom{8}{3}=56\), and all-boy selections are \(\binom{5}{3}=10\). Therefore (56-10=46) ways.
Step 3
Exam Tip
कुल \(\binom{8}{3}=56\) हैं और केवल लड़के \(\binom{5}{3}=10\) हैं। इसलिए (56-10=46) तरीके हैं।
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(12) विद्यार्थियों में से (9) विद्यार्थियों को चुनना (12) में से कितने विद्यार्थियों को न चुनने के बराबर है?
Selecting (9) students from (12) students is equal to not selecting how many students from (12)?
Explanation opens after your attempt
Step 1
Concept
\(\binom{12}{9}=\binom{12}{3}\). Selecting (9) is the same as not selecting (3).
Step 2
Why this answer is correct
The correct answer is B. (3). \(\binom{12}{9}=\binom{12}{3}\). Selecting (9) is the same as not selecting (3).
Step 3
Exam Tip
\(\binom{12}{9}=\binom{12}{3}\) होता है। (9) चुनना (3) न चुनने के बराबर है।
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(11) प्रश्नों में से (2) प्रश्न छोड़ने के कितने तरीके हैं?
In how many ways can (2) questions be left out from (11) questions?
Explanation opens after your attempt
Step 1
Concept
We need to choose (2) questions to leave out. Hence there are \(\binom{11}{2}=55\) ways.
Step 2
Why this answer is correct
The correct answer is B. (55). We need to choose (2) questions to leave out. Hence there are \(\binom{11}{2}=55\) ways.
Step 3
Exam Tip
छोड़ने के लिए (2) प्रश्न चुनने हैं। इसलिए \(\binom{11}{2}=55\) तरीके हैं।
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(7) अध्यायों में से पुनरावृत्ति के लिए ठीक (4) अध्याय चुनने के कितने तरीके हैं?
How many ways are there to choose exactly (4) chapters from (7) chapters for revision?
Explanation opens after your attempt
Step 1
Concept
The order of chapters is not asked. Therefore \(\binom{7}{4}=35\).
Step 2
Why this answer is correct
The correct answer is B. (35). The order of chapters is not asked. Therefore \(\binom{7}{4}=35\).
Step 3
Exam Tip
अध्यायों का क्रम नहीं पूछा गया है। इसलिए \(\binom{7}{4}=35\) होगा।
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(10) अलग-अलग उपहारों में से (0) उपहार या (1) उपहार चुनने के कुल तरीके कितने हैं?
What is the total number of ways to choose (0) gifts or (1) gift from (10) different gifts?
Explanation opens after your attempt
Step 1
Concept
Total ways are \(\binom{10}{0}+\binom{10}{1}=1+10=11\). Include the (0) selection too.
Step 2
Why this answer is correct
The correct answer is B. (11). Total ways are \(\binom{10}{0}+\binom{10}{1}=1+10=11\). Include the (0) selection too.
Step 3
Exam Tip
कुल तरीके \(\binom{10}{0}+\binom{10}{1}=1+10=11\) हैं। (0) चयन को भी शामिल करें।
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(9) रंगों में से (2) रंगों का झंडा बनाना है लेकिन रंगों का स्थान तय नहीं है। कितने चयन हैं?
A flag is to be made by choosing (2) colors from (9) colors, but the positions of colors are not fixed. How many selections are there?
Explanation opens after your attempt
Step 1
Concept
Positions are not fixed, so only colors are selected. The ways are \(\binom{9}{2}=36\).
Step 2
Why this answer is correct
The correct answer is B. (36). Positions are not fixed, so only colors are selected. The ways are \(\binom{9}{2}=36\).
Step 3
Exam Tip
स्थान तय नहीं है इसलिए केवल रंगों का चयन है। तरीके \(\binom{9}{2}=36\) हैं।
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(6) मिठाइयों में से ठीक (2) मिठाइयां चुनने के कितने तरीके हैं?
How many ways are there to choose exactly (2) sweets from (6) sweets?
Explanation opens after your attempt
Step 1
Concept
Choosing exactly (2) means \(\binom{6}{2}\). Its value is (15).
Step 2
Why this answer is correct
The correct answer is B. (15). Choosing exactly (2) means \(\binom{6}{2}\). Its value is (15).
Step 3
Exam Tip
ठीक (2) चुनने का अर्थ \(\binom{6}{2}\) है। इसका मान (15) है।
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(5) विज्ञान और (6) कला विद्यार्थियों में से (1) विज्ञान और (3) कला विद्यार्थी कितने तरीकों से चुने जा सकते हैं?
From (5) science and (6) arts students, in how many ways can (1) science and (3) arts students be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{5}{1}\binom{6}{3}=5\cdot20=100\). Separate the conditions first and then multiply.
Step 2
Why this answer is correct
The correct answer is C. (100). The ways are \(\binom{5}{1}\binom{6}{3}=5\cdot20=100\). Separate the conditions first and then multiply.
Step 3
Exam Tip
तरीके \(\binom{5}{1}\binom{6}{3}=5\cdot20=100\) हैं। पहले शर्तों को अलग करें फिर गुणा करें।
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(4) वरिष्ठ और (5) कनिष्ठ विद्यार्थियों में से (2) वरिष्ठ और (1) कनिष्ठ कितने तरीकों से चुने जा सकते हैं?
From (4) senior and (5) junior students, in how many ways can (2) senior and (1) junior students be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{4}{2}\binom{5}{1}=6\cdot5=30\). Selections from different categories are multiplied.
Step 2
Why this answer is correct
The correct answer is C. (30). The ways are \(\binom{4}{2}\binom{5}{1}=6\cdot5=30\). Selections from different categories are multiplied.
Step 3
Exam Tip
तरीके \(\binom{4}{2}\binom{5}{1}=6\cdot5=30\) हैं। अलग श्रेणियों का चयन गुणा से जुड़ता है।
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(3) हिंदी और (4) अंग्रेजी पुस्तकों में से कुल (2) पुस्तकें चुननी हैं जिनमें दोनों हिंदी हों। कितने तरीके हैं?
From (3) Hindi and (4) English books, (2) books are to be selected and both must be Hindi. How many ways are there?
Explanation opens after your attempt
Step 1
Concept
Both books must be Hindi, so there are \(\binom{3}{2}=3\) ways. Read the condition and choose the correct group.
Step 2
Why this answer is correct
The correct answer is A. (3). Both books must be Hindi, so there are \(\binom{3}{2}=3\) ways. Read the condition and choose the correct group.
Step 3
Exam Tip
दोनों पुस्तकें हिंदी होनी हैं इसलिए \(\binom{3}{2}=3\) तरीके हैं। शर्त पढ़कर सही समूह चुनें।
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(8) अलग-अलग टिकटों में से (5) टिकट चुनने के कितने तरीके हैं?
How many ways are there to choose (5) tickets from (8) different tickets?
Explanation opens after your attempt
Step 1
Concept
\(\binom{8}{5}=\binom{8}{3}=56\). Choosing by the complementary form is easier.
Step 2
Why this answer is correct
The correct answer is B. (56). \(\binom{8}{5}=\binom{8}{3}=56\). Choosing by the complementary form is easier.
Step 3
Exam Tip
\(\binom{8}{5}=\binom{8}{3}=56\) है। पूरक रूप से चुनना आसान है।
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(7) स्वादों में से (3) स्वादों वाली आइसक्रीम कितने तरीकों से चुनी जा सकती है?
In how many ways can an ice cream with (3) flavors be selected from (7) flavors?
Explanation opens after your attempt
Step 1
Concept
The (3) flavors can be selected in \(\binom{7}{3}=35\) ways. The order of flavors is usually not counted.
Step 2
Why this answer is correct
The correct answer is C. (35). The (3) flavors can be selected in \(\binom{7}{3}=35\) ways. The order of flavors is usually not counted.
Step 3
Exam Tip
तीन स्वादों का चयन \(\binom{7}{3}=35\) तरीकों से होगा। स्वादों का क्रम सामान्यतः नहीं गिना जाता।
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(6) चित्रों में से (2) चित्र चुनकर पोस्टर में लगाने हैं। कितने चयन संभव हैं?
(2) pictures are to be selected from (6) pictures for a poster. How many selections are possible?
Explanation opens after your attempt
Step 1
Concept
The ways to choose two pictures are \(\binom{6}{2}=15\). If positions are not given, consider only selection.
Step 2
Why this answer is correct
The correct answer is B. (15). The ways to choose two pictures are \(\binom{6}{2}=15\). If positions are not given, consider only selection.
Step 3
Exam Tip
दो चित्र चुनने के तरीके \(\binom{6}{2}=15\) हैं। लगाने की जगह न दी हो तो केवल चयन मानें।
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(10) अक्षरों में से (4) अक्षरों का चयन कितने तरीकों से हो सकता है?
In how many ways can (4) letters be selected from (10) letters?
Explanation opens after your attempt
Step 1
Concept
Letters are only being selected, so \(\binom{10}{4}=210\). Use combination when arrangement is not asked.
Step 2
Why this answer is correct
The correct answer is C. (210). Letters are only being selected, so \(\binom{10}{4}=210\). Use combination when arrangement is not asked.
Step 3
Exam Tip
अक्षर केवल चुने जा रहे हैं इसलिए \(\binom{10}{4}=210\) होगा। व्यवस्था न पूछी जाए तो संयोजन लगाएं।
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(9) अंकों में से (2) अंक चुनने हैं ताकि अंकों का क्रम महत्वपूर्ण न हो। कितने तरीके हैं?
From (9) digits, (2) digits are to be chosen and the order is not important. How many ways are there?
Explanation opens after your attempt
Step 1
Concept
Order is not important, so \(\binom{9}{2}=36\). This is the key idea of combination.
Step 2
Why this answer is correct
The correct answer is B. (36). Order is not important, so \(\binom{9}{2}=36\). This is the key idea of combination.
Step 3
Exam Tip
क्रम महत्वपूर्ण नहीं है इसलिए \(\binom{9}{2}=36\) है। यही संयोजन की पहचान है।
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(7) विषयों में से अधिकतम (1) विषय चुनने के कितने तरीके हैं?
In how many ways can at most (1) subject be selected from (7) subjects?
Explanation opens after your attempt
Step 1
Concept
At most (1) means selecting (0) or (1). Hence \(\binom{7}{0}+\binom{7}{1}=1+7=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). At most (1) means selecting (0) or (1). Hence \(\binom{7}{0}+\binom{7}{1}=1+7=8\).
Step 3
Exam Tip
अधिकतम (1) का अर्थ (0) या (1) चयन है। इसलिए \(\binom{7}{0}+\binom{7}{1}=1+7=8\) है।
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(4) फलों में से कम से कम (2) फल चुनने के कितने तरीके हैं?
In how many ways can at least (2) fruits be selected from (4) fruits?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{4}{2}+\binom{4}{3}+\binom{4}{4}=6+4+1=11\). Add all possible values in at least questions.
Step 2
Why this answer is correct
The correct answer is C. (11). The ways are \(\binom{4}{2}+\binom{4}{3}+\binom{4}{4}=6+4+1=11\). Add all possible values in at least questions.
Step 3
Exam Tip
तरीके \(\binom{4}{2}+\binom{4}{3}+\binom{4}{4}=6+4+1=11\) हैं। कम से कम में सभी मान जोड़ें।
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