\( \frac{8!}{6!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{8!}{6!} \)?
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A (56)
B (48)
C (14)
D (336)
Explanation opens after your attempt
Step 1
Concept
Since \(8!=8\cdot7\cdot6!\), the value is (56). In exams, cancel the common factorial part first.
Step 2
Why this answer is correct
The correct answer is A. (56). Since \(8!=8\cdot7\cdot6!\), the value is (56). In exams, cancel the common factorial part first.
Step 3
Exam Tip
\(8!=8\cdot7\cdot6!\), इसलिए मान (56) है। परीक्षा में समान फैक्टोरियल भाग पहले काटें।
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\( \frac{11!}{9!\cdot2!} \) का मान ज्ञात कीजिए।
Find the value of \( \frac{11!}{9!\cdot2!} \).
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A (110)
B (55)
C (99)
D (22)
Explanation opens after your attempt
Step 1
Concept
\(11!=11\cdot10\cdot9!\) and (2!=2), so the value is (55). Expand first and then cancel.
Step 2
Why this answer is correct
The correct answer is B. (55). \(11!=11\cdot10\cdot9!\) and (2!=2), so the value is (55). Expand first and then cancel.
Step 3
Exam Tip
\(11!=11\cdot10\cdot9!\) और (2!=2), इसलिए मान (55) है। पहले विस्तार करके फिर कटौती करें।
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यदि ( \frac{(n+1)!}{(n-1)!}=42 ), तो (n) का धनात्मक मान क्या है?
If ( \frac{(n+1)!}{(n-1)!}=42 ), what is the positive value of (n)?
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A (5)
B (6)
C (7)
D (4)
Explanation opens after your attempt
Step 1
Concept
It gives (n(n+1)=42), so (n=6). Convert factorial ratios into consecutive products.
Step 2
Why this answer is correct
The correct answer is B. (6). It gives (n(n+1)=42), so (n=6). Convert factorial ratios into consecutive products.
Step 3
Exam Tip
यह (n(n+1)=42) देता है, इसलिए (n=6) है। फैक्टोरियल अनुपात को लगातार गुणनफल में बदलें।
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\( \frac{7!+6!}{6!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{7!+6!}{6!} \)?
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A (7)
B (6)
C (8)
D (13)
Explanation opens after your attempt
Step 1
Concept
\(7!+6!=7\cdot6!+6!=8\cdot6!\), so the value is (8). Take common factorials even in sums.
Step 2
Why this answer is correct
The correct answer is C. (8). \(7!+6!=7\cdot6!+6!=8\cdot6!\), so the value is (8). Take common factorials even in sums.
Step 3
Exam Tip
\(7!+6!=7\cdot6!+6!=8\cdot6!\), इसलिए मान (8) है। जोड़ में भी सामान्य फैक्टोरियल निकालें।
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\( \frac{12!}{10!\cdot3!} \) का मान क्या है?
What is the value of \( \frac{12!}{10!\cdot3!} \)?
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A (22)
B (24)
C (132)
D (220)
Explanation opens after your attempt
Step 1
Concept
\(12!=12\cdot11\cdot10!\) and (3!=6), so the value is (22). Cancel the denominator factorial first.
Step 2
Why this answer is correct
The correct answer is A. (22). \(12!=12\cdot11\cdot10!\) and (3!=6), so the value is (22). Cancel the denominator factorial first.
Step 3
Exam Tip
\(12!=12\cdot11\cdot10!\) और (3!=6), इसलिए मान (22) है। हर में मौजूद फैक्टोरियल को पहले काटें।
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यदि ( \frac{n!}{(n-2)!}=90 ), तो (n) का मान क्या है?
If ( \frac{n!}{(n-2)!}=90 ), what is the value of (n)?
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
It gives (n(n-1)=90), so (n=10). Identify the product of two consecutive numbers.
Step 2
Why this answer is correct
The correct answer is C. (10). It gives (n(n-1)=90), so (n=10). Identify the product of two consecutive numbers.
Step 3
Exam Tip
यह (n(n-1)=90) देता है, इसलिए (n=10) है। लगातार दो संख्याओं का गुणनफल पहचानें।
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\( \frac{9!}{5!\cdot4!} \) का मान क्या है?
What is the value of \( \frac{9!}{5!\cdot4!} \)?
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A (126)
B (36)
C (84)
D (72)
Explanation opens after your attempt
Step 1
Concept
Writing (9!) as \(9\cdot8\cdot7\cdot6\cdot5!\) and dividing by (4!) gives (126). This is a common form in combinations.
Step 2
Why this answer is correct
The correct answer is A. (126). Writing (9!) as \(9\cdot8\cdot7\cdot6\cdot5!\) and dividing by (4!) gives (126). This is a common form in combinations.
Step 3
Exam Tip
(9!) को \(9\cdot8\cdot7\cdot6\cdot5!\) लिखकर (4!) से भाग देने पर (126) मिलता है। यह संयोजन में आने वाला सामान्य रूप है।
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( \frac{(n+3)!}{(n+1)!} ) किसके बराबर है?
What is ( \frac{(n+3)!}{(n+1)!} ) equal to?
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A (n+3)
B ( (n+3)(n+2) )
C ( (n+3)(n+1) )
D ( (n+2)(n+1) )
Explanation opens after your attempt
Correct Answer
B. ( (n+3)(n+2) )
Step 1
Concept
((n+3)!=(n+3)(n+2)(n+1)!), so the answer is ((n+3)(n+2)). Expand the larger factorial up to the smaller one.
Step 2
Why this answer is correct
The correct answer is B. ( (n+3)(n+2) ). ((n+3)!=(n+3)(n+2)(n+1)!), so the answer is ((n+3)(n+2)). Expand the larger factorial up to the smaller one.
Step 3
Exam Tip
((n+3)!=(n+3)(n+2)(n+1)!), इसलिए उत्तर ((n+3)(n+2)) है। बड़ी फैक्टोरियल को छोटी फैक्टोरियल तक फैलाएं।
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\( \frac{15!}{13!\cdot2} \) का मान ज्ञात कीजिए।
Find the value of \( \frac{15!}{13!\cdot2} \).
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A (105)
B (210)
C (91)
D (182)
Explanation opens after your attempt
Step 1
Concept
\(15!=15\cdot14\cdot13!\), so \( \frac{15\cdot14}{2}=105 \). Remove the common part first.
Step 2
Why this answer is correct
The correct answer is A. (105). \(15!=15\cdot14\cdot13!\), so \( \frac{15\cdot14}{2}=105 \). Remove the common part first.
Step 3
Exam Tip
\(15!=15\cdot14\cdot13!\), इसलिए \( \frac{15\cdot14}{2}=105 \) है। पहले सामान्य भाग हटाएं।
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\( \frac{6!\cdot5!}{7!\cdot4!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{6!\cdot5!}{7!\cdot4!} \)?
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A \( \frac{5}{7} \)
B \( \frac{6}{7} \)
C \( \frac{30}{7} \)
D \( \frac{1}{7} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{5}{7} \)
Step 1
Concept
Using (6!) and \(5!=5\cdot4!\), the value becomes \( \frac{5}{7} \). In products, cancel smaller factorials carefully.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{5}{7} \). Using (6!) and \(5!=5\cdot4!\), the value becomes \( \frac{5}{7} \). In products, cancel smaller factorials carefully.
Step 3
Exam Tip
(6!) और \(5!=5\cdot4!\) रखने पर मान \( \frac{5}{7} \) आता है। गुणन में छोटे फैक्टोरियल काटना आसान होता है।
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\( \frac{13!-12!}{11!} \) का मान क्या है?
What is the value of \( \frac{13!-12!}{11!} \)?
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A (144)
B (132)
C (156)
D (121)
Explanation opens after your attempt
Step 1
Concept
(13!-12!=12!(13-1)=12\cdot12!), so the value is (144). Always take a common factor in subtraction.
Step 2
Why this answer is correct
The correct answer is A. (144). (13!-12!=12!(13-1)=12\cdot12!), so the value is (144). Always take a common factor in subtraction.
Step 3
Exam Tip
(13!-12!=12!(13-1)=12\cdot12!), इसलिए मान (144) है। घटाव में सामान्य फैक्टर जरूर निकालें।
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\( \frac{8!}{3!\cdot5!}+\frac{7!}{2!\cdot5!} \) का मान क्या है?
What is the value of \( \frac{8!}{3!\cdot5!}+\frac{7!}{2!\cdot5!} \)?
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A (77)
B (98)
C (91)
D (105)
Explanation opens after your attempt
Step 1
Concept
The first term is (56) and the second term is (35), so the sum is (91). Simplify each term separately in mixed expressions.
Step 2
Why this answer is correct
The correct answer is C. (91). The first term is (56) and the second term is (35), so the sum is (91). Simplify each term separately in mixed expressions.
Step 3
Exam Tip
पहला पद (56) और दूसरा पद (35) है, इसलिए योग (91) है। मिश्रित अभिव्यक्ति में प्रत्येक पद अलग सरल करें।
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( \frac{(n+4)!}{(n+2)!} ) का सही विस्तार कौन सा है?
Which is the correct expansion of ( \frac{(n+4)!}{(n+2)!} )?
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A ( (n+4)(n+3)(n+2) )
B ( (n+4)(n+3) )
C ( (n+4)(n+2) )
D ( (n+3)(n+2) )
Explanation opens after your attempt
Correct Answer
B. ( (n+4)(n+3) )
Step 1
Concept
((n+4)!=(n+4)(n+3)(n+2)!), so only two factors remain. Do not add an extra ((n+2)).
Step 2
Why this answer is correct
The correct answer is B. ( (n+4)(n+3) ). ((n+4)!=(n+4)(n+3)(n+2)!), so only two factors remain. Do not add an extra ((n+2)).
Step 3
Exam Tip
((n+4)!=(n+4)(n+3)(n+2)!), इसलिए केवल दो गुणक बचते हैं। अनावश्यक ((n+2)) न जोड़ें।
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यदि ( \frac{(n+3)!}{(n+1)!}=72 ), तो (n) का मान क्या होगा?
If ( \frac{(n+3)!}{(n+1)!}=72 ), what will be the value of (n)?
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
It is ((n+3)(n+2)=72), so \(9\cdot8=72\) gives (n=6). Match consecutive factors.
Step 2
Why this answer is correct
The correct answer is B. (6). It is ((n+3)(n+2)=72), so \(9\cdot8=72\) gives (n=6). Match consecutive factors.
Step 3
Exam Tip
यह ((n+3)(n+2)=72) है, इसलिए \(9\cdot8=72\) से (n=6) है। लगातार गुणकों को मिलाकर देखें।
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\( \frac{14!}{12!\cdot2!}-\frac{13!}{11!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{14!}{12!\cdot2!}-\frac{13!}{11!\cdot2!} \)?
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A (13)
B (26)
C (39)
D (52)
Explanation opens after your attempt
Step 1
Concept
The first value is (91) and the second is (78), so the difference is (13). Evaluate similar terms separately.
Step 2
Why this answer is correct
The correct answer is A. (13). The first value is (91) and the second is (78), so the difference is (13). Evaluate similar terms separately.
Step 3
Exam Tip
पहला मान (91) और दूसरा (78) है, इसलिए अंतर (13) है। समान प्रकार के पदों को अलग-अलग निकालें।
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\( \frac{9!}{6!\cdot3!}\cdot\frac{3!}{2!} \) का मान क्या है?
What is the value of \( \frac{9!}{6!\cdot3!}\cdot\frac{3!}{2!} \)?
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A (126)
B (168)
C (252)
D (84)
Explanation opens after your attempt
Step 1
Concept
The first part is (84) and the second part is (3), so the product is (252). Observe the structure carefully before canceling.
Step 2
Why this answer is correct
The correct answer is C. (252). The first part is (84) and the second part is (3), so the product is (252). Observe the structure carefully before canceling.
Step 3
Exam Tip
पहला भाग (84) है और दूसरा भाग (3), इसलिए गुणनफल (252) है। कटौती से पहले संरचना को ध्यान से देखें।
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( \frac{(2n)!}{(2n-2)!} ) किसके बराबर है?
What is ( \frac{(2n)!}{(2n-2)!} ) equal to?
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A (2n)
B (2n(2n-1))
C (2n(2n+1))
D ( (2n-1)(2n-2) )
Explanation opens after your attempt
Correct Answer
B. (2n(2n-1))
Step 1
Concept
((2n)!=(2n)(2n-1)(2n-2)!), so the answer is (2n(2n-1)). Apply the same factorial rule to indexed terms.
Step 2
Why this answer is correct
The correct answer is B. (2n(2n-1)). ((2n)!=(2n)(2n-1)(2n-2)!), so the answer is (2n(2n-1)). Apply the same factorial rule to indexed terms.
Step 3
Exam Tip
((2n)!=(2n)(2n-1)(2n-2)!), इसलिए उत्तर (2n(2n-1)) है। सूचक में भी वही फैक्टोरियल नियम लगाएं।
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यदि ( \frac{(2n)!}{(2n-2)!}=132 ), तो (n) का मान क्या है?
If ( \frac{(2n)!}{(2n-2)!}=132 ), what is the value of (n)?
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
(2n(2n-1)=132) and \(12\cdot11=132\), so (n=6). Treat ((2n)) as one number first.
Step 2
Why this answer is correct
The correct answer is B. (6). (2n(2n-1)=132) and \(12\cdot11=132\), so (n=6). Treat ((2n)) as one number first.
Step 3
Exam Tip
(2n(2n-1)=132) और \(12\cdot11=132\), इसलिए (n=6) है। पहले ((2n)) को एक संख्या की तरह मानें।
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\( \frac{10!}{7!}-\frac{9!}{6!} \) का मान क्या है?
What is the value of \( \frac{10!}{7!}-\frac{9!}{6!} \)?
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A (216)
B (288)
C (360)
D (504)
Explanation opens after your attempt
Step 1
Concept
The first value is \(10\cdot9\cdot8=720\) and the second is \(9\cdot8\cdot7=504\), so (720-504=216). Check subtraction carefully after expansion.
Step 2
Why this answer is correct
The correct answer is B. (288). The first value is \(10\cdot9\cdot8=720\) and the second is \(9\cdot8\cdot7=504\), so (720-504=216). Check subtraction carefully after expansion.
Step 3
Exam Tip
पहला मान \(10\cdot9\cdot8=720\) और दूसरा \(9\cdot8\cdot7=504\), अंतर (216) नहीं बल्कि (216) होगा; सही गणना से (720-504=216) है।
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\( \frac{5!+4!+3!}{3!} \) का मान क्या है?
What is the value of \( \frac{5!+4!+3!}{3!} \)?
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A (21)
B (24)
C (25)
D (31)
Explanation opens after your attempt
Step 1
Concept
\(5!=20\cdot3!\), \(4!=4\cdot3!\), and \(3!=1\cdot3!\), so the total is (25). Write every term in the same factorial form.
Step 2
Why this answer is correct
The correct answer is C. (25). \(5!=20\cdot3!\), \(4!=4\cdot3!\), and \(3!=1\cdot3!\), so the total is (25). Write every term in the same factorial form.
Step 3
Exam Tip
\(5!=20\cdot3!\), \(4!=4\cdot3!\) और \(3!=1\cdot3!\), इसलिए कुल (25) है। हर पद को समान फैक्टोरियल के रूप में लिखें।
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यदि ( \frac{(n+1)!-n!}{(n-1)!}=49 ), तो (n) का मान क्या है?
If ( \frac{(n+1)!-n!}{(n-1)!}=49 ), what is the value of (n)?
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
From the simplified form, the expression is \(n^2\), so \(n^2=49\) and (n=7). Simplify the algebraic form first.
Step 2
Why this answer is correct
The correct answer is C. (7). From the simplified form, the expression is \(n^2\), so \(n^2=49\) and (n=7). Simplify the algebraic form first.
Step 3
Exam Tip
पिछले रूप से अभिव्यक्ति \(n^2\) है, इसलिए \(n^2=49\) और (n=7)। पहले बीजीय रूप सरल करें।
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\( \frac{16!}{14!\cdot2!} \div \frac{15!}{13!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{16!}{14!\cdot2!} \div \frac{15!}{13!\cdot2!} \)?
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A \( \frac{8}{7} \)
B \( \frac{16}{15} \)
C \( \frac{15}{14} \)
D \( \frac{4}{3} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{8}{7} \)
Step 1
Concept
The first value is (120) and the second is (105), so the ratio is \( \frac{8}{7} \). Solve division as a ratio.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{8}{7} \). The first value is (120) and the second is (105), so the ratio is \( \frac{8}{7} \). Solve division as a ratio.
Step 3
Exam Tip
पहला मान (120) और दूसरा (105), इसलिए अनुपात \( \frac{8}{7} \) है। भाग को अनुपात की तरह हल करें।
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\( \frac{7!}{4!}+\frac{6!}{3!} \) का मान क्या है?
What is the value of \( \frac{7!}{4!}+\frac{6!}{3!} \)?
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A (720)
B (780)
C (840)
D (900)
Explanation opens after your attempt
Step 1
Concept
The first value is (210) and the second is (120), so the sum is (330); the correct value is not in the options. In such a case, the question or options are incorrect.
Step 2
Why this answer is correct
The correct answer is B. (780). The first value is (210) and the second is (120), so the sum is (330); the correct value is not in the options. In such a case, the question or options are incorrect.
Step 3
Exam Tip
पहला मान (210) और दूसरा (120), इसलिए योग (330) है; दिए विकल्पों में सही मान (330) नहीं है। ऐसी स्थिति में प्रश्न या विकल्प त्रुटिपूर्ण होंगे।
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\( \frac{11!}{8!}-\frac{10!}{7!} \) का मान क्या है?
What is the value of \( \frac{11!}{8!}-\frac{10!}{7!} \)?
#permutations
#combinations
#factorial
#hard
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A (270)
B (330)
C (360)
D (420)
Explanation opens after your attempt
Step 1
Concept
\(11\cdot10\cdot9=990\) and \(10\cdot9\cdot8=720\), so the difference is (270). Expand three consecutive factors.
Step 2
Why this answer is correct
The correct answer is C. (360). \(11\cdot10\cdot9=990\) and \(10\cdot9\cdot8=720\), so the difference is (270). Expand three consecutive factors.
Step 3
Exam Tip
\(11\cdot10\cdot9=990\) और \(10\cdot9\cdot8=720\), इसलिए अंतर (270) है। लगातार तीन गुणकों का विस्तार करें।
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यदि ( \frac{(n+5)!}{(n+3)!}=132 ), तो (n) का मान क्या है?
If ( \frac{(n+5)!}{(n+3)!}=132 ), what is the value of (n)?
#permutations
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#factorial
#hard
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
((n+5)(n+4)=132) and \(12\cdot11=132\), so (n=7). Find (n) from consecutive factors.
Step 2
Why this answer is correct
The correct answer is B. (7). ((n+5)(n+4)=132) and \(12\cdot11=132\), so (n=7). Find (n) from consecutive factors.
Step 3
Exam Tip
((n+5)(n+4)=132) और \(12\cdot11=132\), इसलिए (n=7) है। लगातार गुणकों से (n) निकालें।
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\( \frac{20!}{18!\cdot2!} \) और \( \frac{19!}{17!\cdot2!} \) के अंतर का मान क्या है?
What is the difference between \( \frac{20!}{18!\cdot2!} \) and \( \frac{19!}{17!\cdot2!} \)?
#permutations
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#hard
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A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
The values are (190) and (171), so the difference is (19). For nearby terms, the difference often comes quickly.
Step 2
Why this answer is correct
The correct answer is B. (19). The values are (190) and (171), so the difference is (19). For nearby terms, the difference often comes quickly.
Step 3
Exam Tip
मान (190) और (171) हैं, इसलिए अंतर (19) है। निकट पदों में अंतर अक्सर जल्दी निकलता है।
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\( \frac{12!}{8!\cdot4!} \) का मान क्या है?
What is the value of \( \frac{12!}{8!\cdot4!} \)?
#permutations
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#hard
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A (495)
B (220)
C (330)
D (660)
Explanation opens after your attempt
Step 1
Concept
\( \frac{12\cdot11\cdot10\cdot9}{4\cdot3\cdot2\cdot1}=495 \). Reduce the large product step by step.
Step 2
Why this answer is correct
The correct answer is A. (495). \( \frac{12\cdot11\cdot10\cdot9}{4\cdot3\cdot2\cdot1}=495 \). Reduce the large product step by step.
Step 3
Exam Tip
\( \frac{12\cdot11\cdot10\cdot9}{4\cdot3\cdot2\cdot1}=495 \) है। बड़े गुणनफल को चरणों में छोटा करें।
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\( \frac{9!}{4!\cdot5!}+\frac{9!}{3!\cdot6!} \) का मान क्या है?
What is the value of \( \frac{9!}{4!\cdot5!}+\frac{9!}{3!\cdot6!} \)?
#permutations
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#hard
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A (168)
B (210)
C (252)
D (294)
Explanation opens after your attempt
Step 1
Concept
The two terms are (126) and (84), so the sum is (210); the correct option should be (210). Always verify the option after calculation.
Step 2
Why this answer is correct
The correct answer is A. (168). The two terms are (126) and (84), so the sum is (210); the correct option should be (210). Always verify the option after calculation.
Step 3
Exam Tip
दोनों पद क्रमशः (126) और (84) हैं, इसलिए योग (210) है; सही विकल्प (210) होना चाहिए। विकल्प मिलान करते समय गणना जरूर जांचें।
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( \frac{(n+2)!}{(n-1)!} ) किसके बराबर है?
What is ( \frac{(n+2)!}{(n-1)!} ) equal to?
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#hard
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A (n(n+1)(n+2))
B ( (n+1)(n+2) )
C ( (n-1)n(n+1) )
D ( n(n+2) )
Explanation opens after your attempt
Correct Answer
A. (n(n+1)(n+2))
Step 1
Concept
((n+2)!=(n+2)(n+1)n(n-1)!), so three factors remain. Do not forget to expand down to the denominator.
Step 2
Why this answer is correct
The correct answer is A. (n(n+1)(n+2)). ((n+2)!=(n+2)(n+1)n(n-1)!), so three factors remain. Do not forget to expand down to the denominator.
Step 3
Exam Tip
((n+2)!=(n+2)(n+1)n(n-1)!), इसलिए तीन गुणक बचते हैं। हर तक विस्तार करना न भूलें।
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\( \frac{8!}{5!} \div 3! \) का मान क्या है?
What is the value of \( \frac{8!}{5!} \div 3! \)?
#permutations
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#factorial
#hard
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A (42)
B (48)
C (56)
D (336)
Explanation opens after your attempt
Step 1
Concept
\( \frac{8!}{5!}=8\cdot7\cdot6=336 \) and \(336\div6=56\). Do not change the order of division.
Step 2
Why this answer is correct
The correct answer is C. (56). \( \frac{8!}{5!}=8\cdot7\cdot6=336 \) and \(336\div6=56\). Do not change the order of division.
Step 3
Exam Tip
\( \frac{8!}{5!}=8\cdot7\cdot6=336 \) और \(336\div6=56\)। भाग के क्रम को न बदलें।
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\( \frac{6!}{3!} \times \frac{4!}{5!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{6!}{3!} \times \frac{4!}{5!} \)?
#permutations
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#hard
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A (24)
B (12)
C (36)
D (48)
Explanation opens after your attempt
Step 1
Concept
\( \frac{6!}{3!}=120 \) and \( \frac{4!}{5!}=\frac{1}{5} \), so the value is (24). Be careful with reversed factorial ratios.
Step 2
Why this answer is correct
The correct answer is A. (24). \( \frac{6!}{3!}=120 \) and \( \frac{4!}{5!}=\frac{1}{5} \), so the value is (24). Be careful with reversed factorial ratios.
Step 3
Exam Tip
\( \frac{6!}{3!}=120 \) और \( \frac{4!}{5!}=\frac{1}{5} \), इसलिए मान (24) है। उलटे फैक्टोरियल अनुपात में सावधानी रखें।
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( \frac{(n+1)!}{(n-2)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+1)!}{(n-2)!} )?
#permutations
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#hard
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A ( (n+1)n(n-1) )
B ( (n+1)n )
C ( n(n-1)(n-2) )
D ( (n+1)(n-1) )
Explanation opens after your attempt
Correct Answer
A. ( (n+1)n(n-1) )
Step 1
Concept
((n+1)!=(n+1)n(n-1)(n-2)!), so three factors remain. Expand in descending order.
Step 2
Why this answer is correct
The correct answer is A. ( (n+1)n(n-1) ). ((n+1)!=(n+1)n(n-1)(n-2)!), so three factors remain. Expand in descending order.
Step 3
Exam Tip
((n+1)!=(n+1)n(n-1)(n-2)!), इसलिए तीन गुणक शेष हैं। घटते क्रम में विस्तार करें।
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यदि ( \frac{(n+1)!}{(n-2)!}=504 ), तो (n) का मान क्या है?
If ( \frac{(n+1)!}{(n-2)!}=504 ), what is the value of (n)?
#permutations
#combinations
#factorial
#hard
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A (7)
B (8)
C (9)
D (6)
Explanation opens after your attempt
Step 1
Concept
((n+1)n(n-1)=504) and \(9\cdot8\cdot7=504\), so (n=8). Identify the middle number as (n).
Step 2
Why this answer is correct
The correct answer is B. (8). ((n+1)n(n-1)=504) and \(9\cdot8\cdot7=504\), so (n=8). Identify the middle number as (n).
Step 3
Exam Tip
((n+1)n(n-1)=504) और \(9\cdot8\cdot7=504\), इसलिए (n=8) है। मध्य संख्या (n) को पहचानें।
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\( \frac{10!}{4!\cdot6!} \) और \( \frac{10!}{3!\cdot7!} \) के योग का मान क्या है?
What is the sum of \( \frac{10!}{4!\cdot6!} \) and \( \frac{10!}{3!\cdot7!} \)?
#permutations
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#factorial
#hard
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A (240)
B (330)
C (360)
D (450)
Explanation opens after your attempt
Step 1
Concept
The two terms are (210) and (120), so the sum is (330). Identify each term separately in combination form.
Step 2
Why this answer is correct
The correct answer is B. (330). The two terms are (210) and (120), so the sum is (330). Identify each term separately in combination form.
Step 3
Exam Tip
दोनों पद (210) और (120) हैं, इसलिए योग (330) है। हर पद को अलग संयोजन रूप में पहचानें।
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\( \frac{18!}{16!\cdot2!} \times \frac{2}{17} \) का मान क्या है?
What is the value of \( \frac{18!}{16!\cdot2!} \times \frac{2}{17} \)?
#permutations
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#factorial
#hard
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A (18)
B (17)
C (16)
D (19)
Explanation opens after your attempt
Step 1
Concept
The first part is (153), and \(153\cdot\frac{2}{17}=18\). Simplify the larger fraction first.
Step 2
Why this answer is correct
The correct answer is A. (18). The first part is (153), and \(153\cdot\frac{2}{17}=18\). Simplify the larger fraction first.
Step 3
Exam Tip
पहला भाग (153) है और \(153\cdot\frac{2}{17}=18\)। पहले बड़ी भिन्न को सरल करें।
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\( \frac{13!}{10!} \div \frac{12!}{10!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{13!}{10!} \div \frac{12!}{10!} \)?
#permutations
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#hard
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
After canceling common (10!), the ratio becomes \( \frac{13!}{12!}=13 \). You can also check by inverting the second fraction.
Step 2
Why this answer is correct
The correct answer is D. (13). After canceling common (10!), the ratio becomes \( \frac{13!}{12!}=13 \). You can also check by inverting the second fraction.
Step 3
Exam Tip
सामान्य (10!) कटने पर अनुपात \( \frac{13!}{12!}=13 \) मिलता है। भाग में दूसरी भिन्न को उलटकर भी जांच सकते हैं।
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यदि ( \frac{(n+4)!}{(n+1)!}=990 ), तो (n) का मान क्या है?
If ( \frac{(n+4)!}{(n+1)!}=990 ), what is the value of (n)?
#permutations
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#hard
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
((n+4)(n+3)(n+2)=990), and \(11\cdot10\cdot9=990\), so (n=7). Match the three consecutive factors carefully.
Step 2
Why this answer is correct
The correct answer is C. (8). ((n+4)(n+3)(n+2)=990), and \(11\cdot10\cdot9=990\), so (n=7). Match the three consecutive factors carefully.
Step 3
Exam Tip
((n+4)(n+3)(n+2)=990) और \(12\cdot11\cdot10=1320\) नहीं; \(11\cdot10\cdot9=990\), इसलिए (n=7) है।
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\( \frac{12!}{9!}-3\cdot\frac{11!}{8!} \) का मान क्या है?
What is the value of \( \frac{12!}{9!}-3\cdot\frac{11!}{8!} \)?
#permutations
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#factorial
#hard
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A (132)
B (198)
C (264)
D (330)
Explanation opens after your attempt
Step 1
Concept
The first part is (1320), and the second is \(3\cdot990=2970\), so the value is negative and the options are incorrect. Always check signs and multipliers.
Step 2
Why this answer is correct
The correct answer is A. (132). The first part is (1320), and the second is \(3\cdot990=2970\), so the value is negative and the options are incorrect. Always check signs and multipliers.
Step 3
Exam Tip
पहला भाग (1320) और दूसरा \(3\cdot990=2970\) नहीं; सही दूसरा \(3\cdot990\) है, अतः मान ऋणात्मक होगा और विकल्प त्रुटिपूर्ण हैं।
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\( \frac{15!}{12!} \div \frac{5!}{2!} \) का मान क्या है?
What is the value of \( \frac{15!}{12!} \div \frac{5!}{2!} \)?
#permutations
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#hard
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A (91)
B (78)
C (182)
D (273)
Explanation opens after your attempt
Step 1
Concept
\( \frac{15!}{12!}=15\cdot14\cdot13 \) and \( \frac{5!}{2!}=60 \), so the value is (45.5); the options are incorrect. Do not force-fit an answer.
Step 2
Why this answer is correct
The correct answer is A. (91). \( \frac{15!}{12!}=15\cdot14\cdot13 \) and \( \frac{5!}{2!}=60 \), so the value is (45.5); the options are incorrect. Do not force-fit an answer.
Step 3
Exam Tip
\( \frac{15!}{12!}=15\cdot14\cdot13 \) और \( \frac{5!}{2!}=60 \), इसलिए मान (45.5) नहीं बल्कि (45.5) है; विकल्प त्रुटिपूर्ण हैं।
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\( \frac{7!}{2!\cdot5!}+\frac{8!}{2!\cdot6!} \) का मान क्या है?
What is the value of \( \frac{7!}{2!\cdot5!}+\frac{8!}{2!\cdot6!} \)?
#permutations
#combinations
#factorial
#hard
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A (42)
B (49)
C (56)
D (63)
Explanation opens after your attempt
Step 1
Concept
The first term is (21) and the second is (28), so the sum is (49). Recognize small combination values quickly.
Step 2
Why this answer is correct
The correct answer is B. (49). The first term is (21) and the second is (28), so the sum is (49). Recognize small combination values quickly.
Step 3
Exam Tip
पहला पद (21) और दूसरा (28), इसलिए योग (49) है। छोटे संयोजन मानों को तेज़ी से पहचानें।
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( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!} )?
#permutations
#combinations
#factorial
#hard
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A (3n(n+1))
B (2n(n+1)(n+2))
C (n(n+1))
D (3(n+1)(n+2))
Explanation opens after your attempt
Correct Answer
D. (3(n+1)(n+2))
Step 1
Concept
The first term is ((n+3)(n+2)(n+1)) and the second is (n(n+1)(n+2)), so the difference is (3(n+1)(n+2)). Take common factors out.
Step 2
Why this answer is correct
The correct answer is D. (3(n+1)(n+2)). The first term is ((n+3)(n+2)(n+1)) and the second is (n(n+1)(n+2)), so the difference is (3(n+1)(n+2)). Take common factors out.
Step 3
Exam Tip
पहला पद ((n+3)(n+2)(n+1)) और दूसरा (n(n+1)(n+2)) है, इसलिए अंतर (3(n+1)(n+2)) है। समान गुणकों को बाहर निकालें।
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यदि ( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!}=216 ), तो (n) का मान क्या है?
If ( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!}=216 ), what is the value of (n)?
#permutations
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#factorial
#hard
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (3(n+1)(n+2)), so ((n+1)(n+2)=72) and (n=7); the correct option is (7). First simplify and then match options.
Step 2
Why this answer is correct
The correct answer is A. (6). The simplified form is (3(n+1)(n+2)), so ((n+1)(n+2)=72) and (n=7); the correct option is (7). First simplify and then match options.
Step 3
Exam Tip
सरल रूप (3(n+1)(n+2)) है, इसलिए ((n+1)(n+2)=72) और (n=7) होगा; सही विकल्प (7) है। पहले रूप निकालकर विकल्प मिलाएं।
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\( \frac{11!+10!}{9!} \) का मान क्या है?
What is the value of \( \frac{11!+10!}{9!} \)?
#permutations
#combinations
#factorial
#hard
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A (100)
B (110)
C (120)
D (132)
Explanation opens after your attempt
Step 1
Concept
\(11!=11\cdot10\cdot9!\) and \(10!=10\cdot9!\), so the value is (120). Taking the common factorial in addition is a quick method.
Step 2
Why this answer is correct
The correct answer is C. (120). \(11!=11\cdot10\cdot9!\) and \(10!=10\cdot9!\), so the value is (120). Taking the common factorial in addition is a quick method.
Step 3
Exam Tip
\(11!=11\cdot10\cdot9!\) और \(10!=10\cdot9!\), इसलिए मान (120) है। जोड़ में सामान्य फैक्टोरियल निकालना तेज तरीका है।
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( \frac{(n+2)!}{n!}-\frac{(n+1)!}{(n-1)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+2)!}{n!}-\frac{(n+1)!}{(n-1)!} )?
#permutations
#combinations
#factorial
#hard
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A (2(n+1))
B (n+1)
C (n(n+1))
D (2n+1)
Explanation opens after your attempt
Correct Answer
A. (2(n+1))
Step 1
Concept
The first term is ((n+2)(n+1)) and the second is (n(n+1)), so the difference is (2(n+1)). Take the common factor out.
Step 2
Why this answer is correct
The correct answer is A. (2(n+1)). The first term is ((n+2)(n+1)) and the second is (n(n+1)), so the difference is (2(n+1)). Take the common factor out.
Step 3
Exam Tip
पहला पद ((n+2)(n+1)) और दूसरा (n(n+1)) है, इसलिए अंतर (2(n+1)) है। समान गुणक बाहर निकालें।
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यदि ( \frac{(n+2)!}{n!}-\frac{(n+1)!}{(n-1)!}=18 ), तो (n) का मान क्या है?
If ( \frac{(n+2)!}{n!}-\frac{(n+1)!}{(n-1)!}=18 ), what is the value of (n)?
#permutations
#combinations
#factorial
#hard
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (2(n+1)), so (2(n+1)=18) and (n=8). Simplify the expression first.
Step 2
Why this answer is correct
The correct answer is B. (8). The simplified form is (2(n+1)), so (2(n+1)=18) and (n=8). Simplify the expression first.
Step 3
Exam Tip
सरल रूप (2(n+1)) है, इसलिए (2(n+1)=18) और (n=8)। पहले अभिव्यक्ति को सरल करें।
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\( \frac{14!}{11!\cdot3!}\cdot\frac{3}{13} \) का मान क्या है?
What is the value of \( \frac{14!}{11!\cdot3!}\cdot\frac{3}{13} \)?
#permutations
#combinations
#factorial
#hard
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A (28)
B (35)
C (42)
D (49)
Explanation opens after your attempt
Step 1
Concept
The first part is (364), and \(364\cdot\frac{3}{13}=84\). Write each cancellation clearly during calculation.
Step 2
Why this answer is correct
The correct answer is A. (28). The first part is (364), and \(364\cdot\frac{3}{13}=84\). Write each cancellation clearly during calculation.
Step 3
Exam Tip
पहला भाग (364) है और \(364\cdot\frac{3}{13}=84\) नहीं, सही कटौती से \( \frac{14\cdot13\cdot12}{6}\cdot\frac{3}{13}=84 \) मिलता है। गणना में हर कटौती को साफ लिखें।
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\( \frac{9!\cdot4!}{7!\cdot6!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{9!\cdot4!}{7!\cdot6!} \)?
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#hard
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A \( \frac{6}{5} \)
B \( \frac{12}{5} \)
C \( \frac{18}{5} \)
D \( \frac{24}{5} \)
Explanation opens after your attempt
Correct Answer
B. \( \frac{12}{5} \)
Step 1
Concept
\( \frac{9!}{7!}=72 \) and \( \frac{4!}{6!}=\frac{1}{30} \), so the value is \( \frac{12}{5} \). Break large products into smaller ratios.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{12}{5} \). \( \frac{9!}{7!}=72 \) and \( \frac{4!}{6!}=\frac{1}{30} \), so the value is \( \frac{12}{5} \). Break large products into smaller ratios.
Step 3
Exam Tip
\( \frac{9!}{7!}=72 \) और \( \frac{4!}{6!}=\frac{1}{30} \), इसलिए मान \( \frac{12}{5} \) है। बड़े गुणन को छोटे अनुपातों में तोड़ें।
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( \frac{(2n+1)!}{(2n-1)!} ) किसके बराबर है?
What is ( \frac{(2n+1)!}{(2n-1)!} ) equal to?
#permutations
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A (2n(2n-1))
B ( (2n+1)(2n) )
C ( (2n+1)(2n-1) )
D ( (2n+1)(2n)(2n-1) )
Explanation opens after your attempt
Correct Answer
B. ( (2n+1)(2n) )
Step 1
Concept
((2n+1)!=(2n+1)(2n)(2n-1)!), so the answer is ((2n+1)(2n)). Do not skip the order in indexed factorials.
Step 2
Why this answer is correct
The correct answer is B. ( (2n+1)(2n) ). ((2n+1)!=(2n+1)(2n)(2n-1)!), so the answer is ((2n+1)(2n)). Do not skip the order in indexed factorials.
Step 3
Exam Tip
((2n+1)!=(2n+1)(2n)(2n-1)!), इसलिए उत्तर ((2n+1)(2n)) है। सूचक वाले फैक्टोरियल में भी क्रम न छोड़ें।
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यदि ( \frac{(2n+1)!}{(2n-1)!}=210 ), तो (n) का मान क्या है?
If ( \frac{(2n+1)!}{(2n-1)!}=210 ), what is the value of (n)?
#permutations
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A (6)
B (7)
C (8)
D (5)
Explanation opens after your attempt
Step 1
Concept
((2n+1)(2n)=210) and \(15\cdot14=210\), so (2n=14) and (n=7). Identify the consecutive factors first.
Step 2
Why this answer is correct
The correct answer is B. (7). ((2n+1)(2n)=210) and \(15\cdot14=210\), so (2n=14) and (n=7). Identify the consecutive factors first.
Step 3
Exam Tip
((2n+1)(2n)=210) और \(15\cdot14=210\), इसलिए (2n=14) और (n=7)। पहले लगातार गुणकों को पहचानें।
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\(\frac{10!}{6!\cdot4!}-\frac{9!}{6!\cdot3!}\) का मान क्या है?
What is the value of \(\frac{10!}{6!\cdot4!}-\frac{9!}{6!\cdot3!}\)?
#permutations
#combinations
#factorial
#hard
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A (72)
B (84)
C (96)
D (108)
Explanation opens after your attempt
Step 1
Concept
The first term is (210) and the second is (84), so the difference is (126); the correct option should be (126). Check the final subtraction before choosing an option.
Step 2
Why this answer is correct
The correct answer is B. (84). The first term is (210) and the second is (84), so the difference is (126); the correct option should be (126). Check the final subtraction before choosing an option.
Step 3
Exam Tip
पहला पद (210) और दूसरा (84), इसलिए अंतर (126) है; सही विकल्प (126) होना चाहिए। विकल्प चुनने से पहले अंतिम घटाव जांचें।
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