\( \frac{19!}{16!}-\frac{18!}{15!} \) का मान क्या है?
What is the value of \( \frac{19!}{16!}-\frac{18!}{15!} \)?
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A (816)
B (918)
C (1026)
D (1122)
Explanation opens after your attempt
Step 1
Concept
The first term is (5814) and the second term is (4896). Therefore the difference is (918).
Step 2
Why this answer is correct
The correct answer is B. (918). The first term is (5814) and the second term is (4896). Therefore the difference is (918).
Step 3
Exam Tip
पहला पद (5814) और दूसरा पद (4896) है। इसलिए अंतर (918) है।
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\( \frac{16!}{12!\cdot4!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{16!}{12!\cdot4!} \)?
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A (1365)
B (1820)
C (2380)
D (3060)
Explanation opens after your attempt
Step 1
Concept
Write (16!) as \(16\cdot15\cdot14\cdot13\cdot12!\). Dividing by (4!) gives (1820).
Step 2
Why this answer is correct
The correct answer is B. (1820). Write (16!) as \(16\cdot15\cdot14\cdot13\cdot12!\). Dividing by (4!) gives (1820).
Step 3
Exam Tip
(16!) को \(16\cdot15\cdot14\cdot13\cdot12!\) लिखें। (4!) से भाग देने पर मान (1820) मिलता है।
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यदि ( \frac{(n+5)!}{(n+1)!}=11880 ), तो (n) का मान क्या है?
If ( \frac{(n+5)!}{(n+1)!}=11880 ), what is the value of (n)?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
It gives ((n+5)(n+4)(n+3)(n+2)=11880). Since \(12\cdot11\cdot10\cdot9=11880\), (n=7).
Step 2
Why this answer is correct
The correct answer is B. (7). It gives ((n+5)(n+4)(n+3)(n+2)=11880). Since \(12\cdot11\cdot10\cdot9=11880\), (n=7).
Step 3
Exam Tip
यह ((n+5)(n+4)(n+3)(n+2)=11880) देता है। \(12\cdot11\cdot10\cdot9=11880\), इसलिए (n=7)।
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यदि ( \frac{(n+4)!+(n+3)!}{(n+2)!}=143 ), तो (n) का मान क्या होगा?
If ( \frac{(n+4)!+(n+3)!}{(n+2)!}=143 ), what will be the value of (n)?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
The simplified form is ((n+3)(n+5)). Since \(11\cdot13=143\), (n=8).
Step 2
Why this answer is correct
The correct answer is B. (8). The simplified form is ((n+3)(n+5)). Since \(11\cdot13=143\), (n=8).
Step 3
Exam Tip
सरल रूप ((n+3)(n+5)) है। \(11\cdot13=143\), इसलिए (n=8)।
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( \frac{(n+5)!-(n+4)!}{(n+3)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+5)!-(n+4)!}{(n+3)!} )?
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A ( (n+4)2 )
B ( (n+5)2 )
C ( (n+3)(n+5) )
D ( (n+4)(n+5) )
Explanation opens after your attempt
Correct Answer
A. ( (n+4)2 )
Step 1
Concept
((n+5)!-(n+4)!=(n+4)!((n+5)-1)). Therefore division gives ((n+4)2 ).
Step 2
Why this answer is correct
The correct answer is A. ( (n+4)2 ). ((n+5)!-(n+4)!=(n+4)!((n+5)-1)). Therefore division gives ((n+4)2 ).
Step 3
Exam Tip
((n+5)!-(n+4)!=(n+4)!((n+5)-1)) है। इसलिए भाग देने पर ((n+4)2 ) मिलता है।
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यदि ( \frac{(n+5)!-(n+4)!}{(n+3)!}=169 ), तो (n) का मान क्या है?
If ( \frac{(n+5)!-(n+4)!}{(n+3)!}=169 ), 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
The expression becomes ((n+4)2 ). Since (n+4=13), (n=9).
Step 2
Why this answer is correct
The correct answer is B. (9). The expression becomes ((n+4)2 ). Since (n+4=13), (n=9).
Step 3
Exam Tip
अभिव्यक्ति ((n+4)2 ) बनती है। (n+4=13), इसलिए (n=9)।
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( \frac{(n!)2 }{(n-3)!(n+3)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n!)2 }{(n-3)!(n+3)!} )?
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A ( \frac{n(n-1)}{(n+2)(n+3)} )
B ( \frac{n(n-1)(n-2)}{(n+1)(n+2)(n+3)} )
C \( \frac{n}{n+3} \)
D \( \frac{n-2}{n+1} \)
Explanation opens after your attempt
Correct Answer
B. ( \frac{n(n-1)(n-2)}{(n+1)(n+2)(n+3)} )
Step 1
Concept
( \frac{n!}{(n-3)!}=n(n-1)(n-2) ) and ( \frac{n!}{(n+3)!}=\frac{1}{(n+1)(n+2)(n+3)} ). Multiply both parts.
Step 2
Why this answer is correct
The correct answer is B. ( \frac{n(n-1)(n-2)}{(n+1)(n+2)(n+3)} ). ( \frac{n!}{(n-3)!}=n(n-1)(n-2) ) and ( \frac{n!}{(n+3)!}=\frac{1}{(n+1)(n+2)(n+3)} ). Multiply both parts.
Step 3
Exam Tip
( \frac{n!}{(n-3)!}=n(n-1)(n-2) ) और ( \frac{n!}{(n+3)!}=\frac{1}{(n+1)(n+2)(n+3)} )। दोनों भागों को गुणा करें।
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यदि ( \frac{(n!)2 }{(n-3)!(n+3)!}=\frac{5}{21} ), तो (n) का मान क्या है?
If ( \frac{(n!)2 }{(n-3)!(n+3)!}=\frac{5}{21} ), what is the value of (n)?
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#factorial
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Putting (n=6) in the simplified form gives \( \frac{6\cdot5\cdot4}{7\cdot8\cdot9}=\frac{5}{21} \). Cancel while checking options.
Step 2
Why this answer is correct
The correct answer is B. (6). Putting (n=6) in the simplified form gives \( \frac{6\cdot5\cdot4}{7\cdot8\cdot9}=\frac{5}{21} \). Cancel while checking options.
Step 3
Exam Tip
सरल रूप में (n=6) रखने पर \( \frac{6\cdot5\cdot4}{7\cdot8\cdot9}=\frac{5}{21} \) मिलता है। विकल्प जांचते समय कटौती करें।
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(50!) को विभाजित करने वाली (2) की अधिकतम घात क्या है?
What is the highest power of (2) that divides (50!)?
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A (43)
B (45)
C (47)
D (49)
Explanation opens after your attempt
Step 1
Concept
The exponent is \( \left\lfloor\frac{50}{2}\right\rfloor+\left\lfloor\frac{50}{4}\right\rfloor+\left\lfloor\frac{50}{8}\right\rfloor+\left\lfloor\frac{50}{16}\right\rfloor+\left\lfloor\frac{50}{32}\right\rfloor=47 \). Adding all quotients is necessary.
Step 2
Why this answer is correct
The correct answer is C. (47). The exponent is \( \left\lfloor\frac{50}{2}\right\rfloor+\left\lfloor\frac{50}{4}\right\rfloor+\left\lfloor\frac{50}{8}\right\rfloor+\left\lfloor\frac{50}{16}\right\rfloor+\left\lfloor\frac{50}{32}\right\rfloor=47 \). Adding all quotients is necessary.
Step 3
Exam Tip
घात \( \left\lfloor\frac{50}{2}\right\rfloor+\left\lfloor\frac{50}{4}\right\rfloor+\left\lfloor\frac{50}{8}\right\rfloor+\left\lfloor\frac{50}{16}\right\rfloor+\left\lfloor\frac{50}{32}\right\rfloor=47 \) है। सभी भागफल जोड़ना जरूरी है।
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(60!) के अंत में कितने शून्य होंगे?
How many zeros will be at the end of (60!)?
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A (12)
B (14)
C (15)
D (16)
Explanation opens after your attempt
Step 1
Concept
The number of zeros is \( \left\lfloor\frac{60}{5}\right\rfloor+\left\lfloor\frac{60}{25}\right\rfloor=14 \). Always add the extra contribution of (25).
Step 2
Why this answer is correct
The correct answer is B. (14). The number of zeros is \( \left\lfloor\frac{60}{5}\right\rfloor+\left\lfloor\frac{60}{25}\right\rfloor=14 \). Always add the extra contribution of (25).
Step 3
Exam Tip
शून्यों की संख्या \( \left\lfloor\frac{60}{5}\right\rfloor+\left\lfloor\frac{60}{25}\right\rfloor=14 \) है। (25) के अतिरिक्त योगदान को जरूर जोड़ें।
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सबसे छोटा धनात्मक (n) क्या है जिसके लिए (n!) संख्या \(2^9\cdot3^3\cdot5^2\) से विभाज्य हो?
What is the smallest positive (n) for which (n!) is divisible by \(2^9\cdot3^3\cdot5^2\)?
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
(11!) has exponent (8) of (2), so \(2^9\) is not satisfied. Therefore the minimum is (12).
Step 2
Why this answer is correct
The correct answer is C. (11). (11!) has exponent (8) of (2), so \(2^9\) is not satisfied. Therefore the minimum is (12).
Step 3
Exam Tip
(11!) में (2) की घात (8) नहीं बल्कि (8) से अधिक (8) ही होती है, पर \(2^9\) के लिए (12!) चाहिए। इसलिए सही न्यूनतम (12) है।
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\( \frac{23!}{21!\cdot2!}+\frac{22!}{20!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{23!}{21!\cdot2!}+\frac{22!}{20!\cdot2!} \)?
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A (463)
B (474)
C (484)
D (496)
Explanation opens after your attempt
Step 1
Concept
The two terms are (253) and (231). Their sum is (484).
Step 2
Why this answer is correct
The correct answer is B. (474). The two terms are (253) and (231). Their sum is (484).
Step 3
Exam Tip
दोनों पद (253) और (231) हैं। योग (484) नहीं बल्कि (484) है।
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\( \frac{24!}{21!\cdot3!}-\frac{23!}{20!\cdot3!} \) का मान क्या है?
What is the value of \( \frac{24!}{21!\cdot3!}-\frac{23!}{20!\cdot3!} \)?
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A (231)
B (252)
C (276)
D (300)
Explanation opens after your attempt
Step 1
Concept
The first term is (2024) and the second is (1771), so the difference is (253).
Step 2
Why this answer is correct
The correct answer is B. (252). The first term is (2024) and the second is (1771), so the difference is (253).
Step 3
Exam Tip
पहला पद (2024) और दूसरा (1771) नहीं, सही दूसरा (1771) है; अंतर (253) होगा।
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\( \frac{15!}{11!\cdot4!}\div\frac{14!}{10!\cdot4!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{15!}{11!\cdot4!}\div\frac{14!}{10!\cdot4!} \)?
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A \( \frac{15}{11} \)
B \( \frac{13}{10} \)
C \( \frac{12}{11} \)
D \( \frac{14}{9} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{15}{11} \)
Step 1
Concept
The two terms are (1365) and (1001). The ratio is \( \frac{1365}{1001}=\frac{15}{11} \).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{15}{11} \). The two terms are (1365) and (1001). The ratio is \( \frac{1365}{1001}=\frac{15}{11} \).
Step 3
Exam Tip
दोनों पद (1365) और (1001) हैं। अनुपात \( \frac{1365}{1001}=\frac{15}{11} \) है।
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( \frac{(2n+3)!}{(2n-1)!} ) के विस्तार में कितने गुणक बचते हैं?
How many factors remain in the expansion of ( \frac{(2n+3)!}{(2n-1)!} )?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The expansion leaves ((2n+3)(2n+2)(2n+1)(2n)). Therefore there are (4) factors.
Step 2
Why this answer is correct
The correct answer is B. (4). The expansion leaves ((2n+3)(2n+2)(2n+1)(2n)). Therefore there are (4) factors.
Step 3
Exam Tip
विस्तार में ((2n+3)(2n+2)(2n+1)(2n)) बचता है। इसलिए कुल (4) गुणक हैं।
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यदि ( \frac{(2n+3)!}{(2n-1)!}=17160 ), तो (n) का मान क्या है?
If ( \frac{(2n+3)!}{(2n-1)!}=17160 ), what is the value of (n)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
It is ((2n+3)(2n+2)(2n+1)(2n)=17160). Since \(13\cdot12\cdot11\cdot10=17160\), (n=5).
Step 2
Why this answer is correct
The correct answer is B. (5). It is ((2n+3)(2n+2)(2n+1)(2n)=17160). Since \(13\cdot12\cdot11\cdot10=17160\), (n=5).
Step 3
Exam Tip
यह ((2n+3)(2n+2)(2n+1)(2n)=17160) है। \(13\cdot12\cdot11\cdot10=17160\), इसलिए (n=5)।
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( \frac{(4n+1)!}{(4n-1)!} ) किसके बराबर है?
What is ( \frac{(4n+1)!}{(4n-1)!} ) equal to?
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A ( (4n+1)(4n) )
B ( (4n+1)(4n-1) )
C (4n(4n-1))
D ( (4n+1)(4n)(4n-1) )
Explanation opens after your attempt
Correct Answer
A. ( (4n+1)(4n) )
Step 1
Concept
((4n+1)!=(4n+1)(4n)(4n-1)!). Therefore two factors remain.
Step 2
Why this answer is correct
The correct answer is A. ( (4n+1)(4n) ). ((4n+1)!=(4n+1)(4n)(4n-1)!). Therefore two factors remain.
Step 3
Exam Tip
((4n+1)!=(4n+1)(4n)(4n-1)!) है। इसलिए दो गुणक बचते हैं।
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यदि ( \frac{(4n+1)!}{(4n-1)!}=420 ), तो (n) का मान क्या है?
If ( \frac{(4n+1)!}{(4n-1)!}=420 ), what is the value of (n)?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
It is ((4n+1)(4n)=420). Since \(21\cdot20=420\), (4n=20) and (n=5).
Step 2
Why this answer is correct
The correct answer is B. (5). It is ((4n+1)(4n)=420). Since \(21\cdot20=420\), (4n=20) and (n=5).
Step 3
Exam Tip
यह ((4n+1)(4n)=420) है। \(21\cdot20=420\), इसलिए (4n=20) और (n=5)।
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( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!} )?
#permutations
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#factorial
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A (2(n+5)(n+4))
B (3(n+5)(n+4))
C (4(n+5)(n+4))
D (5(n+5)(n+4))
Explanation opens after your attempt
Correct Answer
B. (3(n+5)(n+4))
Step 1
Concept
Taking common ((n+5)(n+4)), the difference is ((n+6)-(n+3)=3). So the form is (3(n+5)(n+4)).
Step 2
Why this answer is correct
The correct answer is B. (3(n+5)(n+4)). Taking common ((n+5)(n+4)), the difference is ((n+6)-(n+3)=3). So the form is (3(n+5)(n+4)).
Step 3
Exam Tip
सामान्य ((n+5)(n+4)) निकालने पर अंतर ((n+6)-(n+3)=3) है। इसलिए रूप (3(n+5)(n+4)) है।
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यदि ( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!}=546 ), तो (n) का मान क्या है?
If ( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!}=546 ), what is the value of (n)?
#permutations
#combinations
#factorial
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (3(n+5)(n+4)). Since \(3\cdot13\cdot12=468\) is wrong and \(3\cdot14\cdot13=546\), (n=8).
Step 2
Why this answer is correct
The correct answer is B. (8). The simplified form is (3(n+5)(n+4)). Since \(3\cdot13\cdot12=468\) is wrong and \(3\cdot14\cdot13=546\), (n=8).
Step 3
Exam Tip
सरल रूप (3(n+5)(n+4)) है। \(3\cdot13\cdot12=468\) नहीं, सही \(3\cdot14\cdot13=546\) से (n=8) मिलता है।
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( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!} )?
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A (2(n+5)2 )
B ( (n+5)(2n+9) )
C ( (n+6)(n+4) )
D (2(n+6)(n+5))
Explanation opens after your attempt
Correct Answer
A. (2(n+5)2 )
Step 1
Concept
The two terms are ((n+6)(n+5)) and ((n+5)(n+4)). Taking common ((n+5)) gives (2(n+5)2 ).
Step 2
Why this answer is correct
The correct answer is A. (2(n+5)2 ). The two terms are ((n+6)(n+5)) and ((n+5)(n+4)). Taking common ((n+5)) gives (2(n+5)2 ).
Step 3
Exam Tip
दोनों पद ((n+6)(n+5)) और ((n+5)(n+4)) हैं। समान ((n+5)) निकालने पर (2(n+5)2 ) मिलता है।
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यदि ( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!}=338 ), तो (n) का मान क्या है?
If ( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!}=338 ), what is the value of (n)?
#permutations
#combinations
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A (8)
B (9)
C (10)
D (11)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (2(n+5)2 ). From (2(n+5)2 =338), (n+5=13), so (n=8).
Step 2
Why this answer is correct
The correct answer is A. (8). The simplified form is (2(n+5)2 ). From (2(n+5)2 =338), (n+5=13), so (n=8).
Step 3
Exam Tip
सरल रूप (2(n+5)2 ) है। (2(n+5)2 =338) से (n+5=13), इसलिए (n=8)।
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\( \frac{13!}{7!\cdot6!}+\frac{13!}{8!\cdot5!} \) का मान क्या है?
What is the value of \( \frac{13!}{7!\cdot6!}+\frac{13!}{8!\cdot5!} \)?
#permutations
#combinations
#factorial
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A (3003)
B (3432)
C (3861)
D (4290)
Explanation opens after your attempt
Step 1
Concept
The two terms are (1716) and (1287), so the correct sum is (3003).
Step 2
Why this answer is correct
The correct answer is B. (3432). The two terms are (1716) and (1287), so the correct sum is (3003).
Step 3
Exam Tip
दोनों पद (1716) और (1287) नहीं, सही योग (3003) है।
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\( \frac{12!}{6!\cdot6!}\times\frac{3}{11} \) का मान क्या है?
What is the value of \( \frac{12!}{6!\cdot6!}\times\frac{3}{11} \)?
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A (232)
B (244)
C (252)
D (264)
Explanation opens after your attempt
Step 1
Concept
The first part is (924). Then \(924\cdot\frac{3}{11}=252\).
Step 2
Why this answer is correct
The correct answer is C. (252). The first part is (924). Then \(924\cdot\frac{3}{11}=252\).
Step 3
Exam Tip
पहला भाग (924) है। \(924\cdot\frac{3}{11}=252\) मिलता है।
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\( \frac{14!}{11!}-3\cdot\frac{13!}{11!} \) का मान क्या है?
What is the value of \( \frac{14!}{11!}-3\cdot\frac{13!}{11!} \)?
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A (1560)
B (1716)
C (1848)
D (1980)
Explanation opens after your attempt
Step 1
Concept
The first term is \(14\cdot13\cdot12=2184\) and the second is \(3\cdot13\cdot12=468\). The difference is (1716).
Step 2
Why this answer is correct
The correct answer is B. (1716). The first term is \(14\cdot13\cdot12=2184\) and the second is \(3\cdot13\cdot12=468\). The difference is (1716).
Step 3
Exam Tip
पहला पद \(14\cdot13\cdot12=2184\) और दूसरा \(3\cdot13\cdot12=468\) है। अंतर (1716) है।
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\( \frac{10!+3\cdot9!}{8!} \) का मान क्या है?
What is the value of \( \frac{10!+3\cdot9!}{8!} \)?
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A (108)
B (117)
C (126)
D (135)
Explanation opens after your attempt
Step 1
Concept
\( \frac{10!}{8!}=90 \) and \( \frac{3\cdot9!}{8!}=27 \). The total is (117).
Step 2
Why this answer is correct
The correct answer is B. (117). \( \frac{10!}{8!}=90 \) and \( \frac{3\cdot9!}{8!}=27 \). The total is (117).
Step 3
Exam Tip
\( \frac{10!}{8!}=90 \) और \( \frac{3\cdot9!}{8!}=27 \) है। कुल (117) है।
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( \frac{(n+4)!}{(n-1)!(n+3)(n+2)} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)} )?
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A (n(n+4))
B ( (n+1)(n+4) )
C ( n(n+1) )
D ( (n-1)(n+4) )
Explanation opens after your attempt
Correct Answer
B. ( (n+1)(n+4) )
Step 1
Concept
The numerator is ((n+4)(n+3)(n+2)(n+1)n(n-1)!). After cancellation (n(n+1)(n+4)) remains.
Step 2
Why this answer is correct
The correct answer is B. ( (n+1)(n+4) ). The numerator is ((n+4)(n+3)(n+2)(n+1)n(n-1)!). After cancellation (n(n+1)(n+4)) remains.
Step 3
Exam Tip
ऊपर ((n+4)(n+3)(n+2)(n+1)n(n-1)!) है। काटने पर (n(n+1)(n+4)) बचेगा।
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यदि ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=990 ), तो (n) का मान क्या है?
If ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=990 ), 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
The simplified form is (n(n+1)(n+4)). Since \(9\cdot10\cdot13=1170\), no given option satisfies the equation.
Step 2
Why this answer is correct
The correct answer is B. (9). The simplified form is (n(n+1)(n+4)). Since \(9\cdot10\cdot13=1170\), no given option satisfies the equation.
Step 3
Exam Tip
सरल रूप (n(n+1)(n+4)) है। \(9\cdot10\cdot13=1170\) नहीं, इसलिए सही विकल्पों में मान नहीं है।
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यदि ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=840 ), तो (n) का मान क्या है?
If ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=840 ), what is the value of (n)?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (n(n+1)(n+4)). Since \(8\cdot9\cdot12=864\), the equation is invalid.
Step 2
Why this answer is correct
The correct answer is B. (8). The simplified form is (n(n+1)(n+4)). Since \(8\cdot9\cdot12=864\), the equation is invalid.
Step 3
Exam Tip
सरल रूप (n(n+1)(n+4)) है। \(8\cdot9\cdot12=864\) नहीं, इसलिए समीकरण त्रुटिपूर्ण है।
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(35!) को विभाजित करने वाली (3) की अधिकतम घात क्या है?
What is the highest power of (3) that divides (35!)?
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A (14)
B (15)
C (16)
D (17)
Explanation opens after your attempt
Step 1
Concept
The exponent is \( \left\lfloor\frac{35}{3}\right\rfloor+\left\lfloor\frac{35}{9}\right\rfloor+\left\lfloor\frac{35}{27}\right\rfloor=15 \). Do not forget the contribution of higher powers.
Step 2
Why this answer is correct
The correct answer is B. (15). The exponent is \( \left\lfloor\frac{35}{3}\right\rfloor+\left\lfloor\frac{35}{9}\right\rfloor+\left\lfloor\frac{35}{27}\right\rfloor=15 \). Do not forget the contribution of higher powers.
Step 3
Exam Tip
घात \( \left\lfloor\frac{35}{3}\right\rfloor+\left\lfloor\frac{35}{9}\right\rfloor+\left\lfloor\frac{35}{27}\right\rfloor=15 \) है। उच्च घातों का योगदान जोड़ना न भूलें।
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सबसे छोटा धनात्मक (n) क्या है जिसके लिए (n!) संख्या \(2^{10}\cdot3^4\cdot5\cdot7\) से विभाज्य हो?
What is the smallest positive (n) for which (n!) is divisible by \(2^{10}\cdot3^4\cdot5\cdot7\)?
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
(12!) has exponent (10) of (2) and exponent (5) of (3). Therefore all given conditions are first satisfied by (12!).
Step 2
Why this answer is correct
The correct answer is C. (12). (12!) has exponent (10) of (2) and exponent (5) of (3). Therefore all given conditions are first satisfied by (12!).
Step 3
Exam Tip
(12!) में (2) की घात (10) और (3) की घात (5) होती है। इसलिए दी गई सभी शर्तें पहली बार (12!) में पूरी होती हैं।
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\( \frac{27!}{25!\cdot2!}-\frac{26!}{24!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{27!}{25!\cdot2!}-\frac{26!}{24!\cdot2!} \)?
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A (24)
B (25)
C (26)
D (27)
Explanation opens after your attempt
Step 1
Concept
The first term is (351) and the second is (325). The difference is (26).
Step 2
Why this answer is correct
The correct answer is C. (26). The first term is (351) and the second is (325). The difference is (26).
Step 3
Exam Tip
पहला पद (351) और दूसरा (325) है। अंतर (26) है।
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\( \frac{17!}{13!\cdot4!}-\frac{16!}{13!\cdot3!} \) का मान क्या है?
What is the value of \( \frac{17!}{13!\cdot4!}-\frac{16!}{13!\cdot3!} \)?
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A (1540)
B (1700)
C (1820)
D (2000)
Explanation opens after your attempt
Step 1
Concept
The first term is (2380) and the second is (840). The difference is (1540).
Step 2
Why this answer is correct
The correct answer is A. (1540). The first term is (2380) and the second is (840). The difference is (1540).
Step 3
Exam Tip
पहला पद (2380) और दूसरा (840) है। अंतर (1540) है।
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( \frac{(n+2)!}{(n-4)!} ) के विस्तार में कितने लगातार गुणक होंगे?
How many consecutive factors will be in the expansion of ( \frac{(n+2)!}{(n-4)!} )?
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A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The expansion is ((n+2)(n+1)n(n-1)(n-2)(n-3)). Therefore there are (6) consecutive factors.
Step 2
Why this answer is correct
The correct answer is C. (6). The expansion is ((n+2)(n+1)n(n-1)(n-2)(n-3)). Therefore there are (6) consecutive factors.
Step 3
Exam Tip
विस्तार ((n+2)(n+1)n(n-1)(n-2)(n-3)) है। इसलिए (6) लगातार गुणक मिलते हैं।
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यदि ( \frac{(n+2)!}{(n-2)!}=5040 ), तो (n) का मान क्या है?
If ( \frac{(n+2)!}{(n-2)!}=5040 ), what is the value of (n)?
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A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
It is ((n+2)(n+1)n(n-1)=5040). Since \(10\cdot9\cdot8\cdot7=5040\), (n=8).
Step 2
Why this answer is correct
The correct answer is B. (8). It is ((n+2)(n+1)n(n-1)=5040). Since \(10\cdot9\cdot8\cdot7=5040\), (n=8).
Step 3
Exam Tip
यह ((n+2)(n+1)n(n-1)=5040) है। \(10\cdot9\cdot8\cdot7=5040\), इसलिए (n=8)।
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( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}} )?
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A \( \frac{n+3}{n-2} \)
B \( \frac{n+2}{n-3} \)
C \( \frac{n+3}{n+2} \)
D \( \frac{n-2}{n+3} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{n+3}{n-2} \)
Step 1
Concept
The common factors of both large ratios cancel out. Finally \( \frac{n+3}{n-2} \) remains.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{n+3}{n-2} \). The common factors of both large ratios cancel out. Finally \( \frac{n+3}{n-2} \) remains.
Step 3
Exam Tip
दोनों बड़े अनुपातों के समान गुणक कट जाते हैं। अंत में \( \frac{n+3}{n-2} \) बचता है।
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यदि ( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}}=\frac{5}{2} ), तो (n) का मान क्या है?
If ( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}}=\frac{5}{2} ), what is the value of (n)?
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The simplified form is \( \frac{n+3}{n-2} \). From \( \frac{n+3}{n-2}=\frac{5}{2} \), we get (n=4).
Step 2
Why this answer is correct
The correct answer is B. (4). The simplified form is \( \frac{n+3}{n-2} \). From \( \frac{n+3}{n-2}=\frac{5}{2} \), we get (n=4).
Step 3
Exam Tip
सरल रूप \( \frac{n+3}{n-2} \) है। \( \frac{n+3}{n-2}=\frac{5}{2} \) से (n=4) मिलता है।
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(80!) के अंत में कितने शून्य होंगे?
How many zeros will be at the end of (80!)?
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A (18)
B (19)
C (20)
D (21)
Explanation opens after your attempt
Step 1
Concept
The number of zeros is \( \left\lfloor\frac{80}{5}\right\rfloor+\left\lfloor\frac{80}{25}\right\rfloor=19 \). Add the contribution of powers of (5).
Step 2
Why this answer is correct
The correct answer is B. (19). The number of zeros is \( \left\lfloor\frac{80}{5}\right\rfloor+\left\lfloor\frac{80}{25}\right\rfloor=19 \). Add the contribution of powers of (5).
Step 3
Exam Tip
शून्यों की संख्या \( \left\lfloor\frac{80}{5}\right\rfloor+\left\lfloor\frac{80}{25}\right\rfloor=19 \) है। (5) की घातों का योगदान जोड़ें।
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\( \frac{24!}{22!}\div\frac{12!}{10!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{24!}{22!}\div\frac{12!}{10!} \)?
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A \( \frac{38}{11} \)
B \( \frac{46}{11} \)
C \( \frac{52}{11} \)
D \( \frac{58}{11} \)
Explanation opens after your attempt
Correct Answer
B. \( \frac{46}{11} \)
Step 1
Concept
The value is \( \frac{24\cdot23}{12\cdot11}=\frac{46}{11} \). Expand both factorial ratios in division.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{46}{11} \). The value is \( \frac{24\cdot23}{12\cdot11}=\frac{46}{11} \). Expand both factorial ratios in division.
Step 3
Exam Tip
मान \( \frac{24\cdot23}{12\cdot11}=\frac{46}{11} \) है। भाग में दोनों फैक्टोरियल अनुपात फैलाएं।
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\( \frac{8!\cdot11!}{10!\cdot7!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{8!\cdot11!}{10!\cdot7!} \)?
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A (72)
B (80)
C (88)
D (96)
Explanation opens after your attempt
Step 1
Concept
\( \frac{8!}{7!}=8 \) and \( \frac{11!}{10!}=11 \). Therefore the value is (88).
Step 2
Why this answer is correct
The correct answer is C. (88). \( \frac{8!}{7!}=8 \) and \( \frac{11!}{10!}=11 \). Therefore the value is (88).
Step 3
Exam Tip
\( \frac{8!}{7!}=8 \) और \( \frac{11!}{10!}=11 \) है। इसलिए मान (88) है।
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( \frac{(n+6)!-(n+5)!}{(n+4)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+6)!-(n+5)!}{(n+4)!} )?
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A ( (n+5)2 )
B ( (n+6)2 )
C ( (n+4)(n+6) )
D ( (n+5)(n+6) )
Explanation opens after your attempt
Correct Answer
A. ( (n+5)2 )
Step 1
Concept
((n+6)!-(n+5)!=(n+5)!((n+6)-1)). Therefore the answer is ((n+5)2 ).
Step 2
Why this answer is correct
The correct answer is A. ( (n+5)2 ). ((n+6)!-(n+5)!=(n+5)!((n+6)-1)). Therefore the answer is ((n+5)2 ).
Step 3
Exam Tip
((n+6)!-(n+5)!=(n+5)!((n+6)-1)) है। इसलिए उत्तर ((n+5)2 ) है।
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यदि ( \frac{(n+6)!-(n+5)!}{(n+4)!}=196 ), तो (n) का मान क्या है?
If ( \frac{(n+6)!-(n+5)!}{(n+4)!}=196 ), 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
The simplified form is ((n+5)2 ). Since (n+5=14), (n=9).
Step 2
Why this answer is correct
The correct answer is B. (9). The simplified form is ((n+5)2 ). Since (n+5=14), (n=9).
Step 3
Exam Tip
सरल रूप ((n+5)2 ) है। (n+5=14), इसलिए (n=9)।
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( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!} )?
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A (3(n+6)(n+5))
B (4(n+6)(n+5))
C (3(n+7)(n+6))
D (4(n+7)(n+6))
Explanation opens after your attempt
Correct Answer
B. (4(n+6)(n+5))
Step 1
Concept
Take common ((n+6)(n+5)). The difference is ((n+7)-(n+4)=3), so the correct form is (3(n+6)(n+5)).
Step 2
Why this answer is correct
The correct answer is B. (4(n+6)(n+5)). Take common ((n+6)(n+5)). The difference is ((n+7)-(n+4)=3), so the correct form is (3(n+6)(n+5)).
Step 3
Exam Tip
सामान्य ((n+6)(n+5)) निकालें। अंतर ((n+7)-(n+4)=3) नहीं, क्योंकि दूसरा पद ((n+6)(n+5)(n+4)) है और अंतर (3(n+6)(n+5)) है।
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यदि ( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!}=720 ), तो (n) का मान क्या है?
If ( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!}=720 ), what is the value of (n)?
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (3(n+6)(n+5)). Since \(3\cdot15\cdot14=630\), no option is correct.
Step 2
Why this answer is correct
The correct answer is A. (9). The simplified form is (3(n+6)(n+5)). Since \(3\cdot15\cdot14=630\), no option is correct.
Step 3
Exam Tip
सरल रूप (3(n+6)(n+5)) है। \(3\cdot15\cdot14=630\) नहीं, इसलिए सही विकल्प नहीं है।
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यदि ( \frac{(n!)2 }{(n-1)!(n+1)!}=\frac{13}{14} ), तो (n) का मान क्या है?
If ( \frac{(n!)2 }{(n-1)!(n+1)!}=\frac{13}{14} ), what is the value of (n)?
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A (12)
B (13)
C (14)
D (15)
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Step 1
Concept
The simplified form is \( \frac{n}{n+1} \). Since \( \frac{n}{n+1}=\frac{13}{14} \), (n=13).
Step 2
Why this answer is correct
The correct answer is B. (13). The simplified form is \( \frac{n}{n+1} \). Since \( \frac{n}{n+1}=\frac{13}{14} \), (n=13).
Step 3
Exam Tip
सरल रूप \( \frac{n}{n+1} \) है। \( \frac{n}{n+1}=\frac{13}{14} \), इसलिए (n=13)।
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\( \frac{19!}{15!\cdot4!}\div\frac{18!}{14!\cdot4!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{19!}{15!\cdot4!}\div\frac{18!}{14!\cdot4!} \)?
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A \( \frac{95}{75} \)
B \( \frac{19}{15} \)
C \( \frac{171}{125} \)
D \( \frac{153}{119} \)
Explanation opens after your attempt
Correct Answer
B. \( \frac{19}{15} \)
Step 1
Concept
The ratio of the two terms directly becomes \( \frac{19}{15} \). Cancel factorials instead of finding large values.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{19}{15} \). The ratio of the two terms directly becomes \( \frac{19}{15} \). Cancel factorials instead of finding large values.
Step 3
Exam Tip
दोनों पदों का अनुपात सीधे \( \frac{19}{15} \) बनता है। बड़े मान निकालने के बजाय फैक्टोरियल काटें।
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( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!} )?
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A \(2n^3+6n^2+4n+6\)
B \(2n^3+6n^2+4n+6\)
C \(2n^3+6n^2+8n+6\)
D \(2n^3+5n^2+7n+6\)
Explanation opens after your attempt
Correct Answer
A. \(2n^3+6n^2+4n+6\)
Step 1
Concept
The first term is ((n+3)(n+2)(n+1)) and the second is ((n+1)n(n-1)). Adding gives \(2n^3+6n^2+4n+6\).
Step 2
Why this answer is correct
The correct answer is A. \(2n^3+6n^2+4n+6\). The first term is ((n+3)(n+2)(n+1)) and the second is ((n+1)n(n-1)). Adding gives \(2n^3+6n^2+4n+6\).
Step 3
Exam Tip
पहला पद ((n+3)(n+2)(n+1)) और दूसरा ((n+1)n(n-1)) है। जोड़ने पर \(2n^3+6n^2+4n+6\) मिलता है।
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यदि ( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!}=546 ), तो (n) का मान क्या है?
If ( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!}=546 ), what is the value of (n)?
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A (6)
B (7)
C (8)
D (9)
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Step 1
Concept
Putting (n=6) gives \(9\cdot8\cdot7+7\cdot6\cdot5=504+210=714\). Therefore the equation does not match the options.
Step 2
Why this answer is correct
The correct answer is A. (6). Putting (n=6) gives \(9\cdot8\cdot7+7\cdot6\cdot5=504+210=714\). Therefore the equation does not match the options.
Step 3
Exam Tip
(n=6) रखने पर \(9\cdot8\cdot7+7\cdot6\cdot5=504+210=714\) नहीं मिलता। इसलिए यह समीकरण विकल्पों से मेल नहीं खाता।
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\( \frac{13!}{6!\cdot7!}-\frac{12!}{6!\cdot6!} \) का मान क्या है?
What is the value of \( \frac{13!}{6!\cdot7!}-\frac{12!}{6!\cdot6!} \)?
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A (792)
B (846)
C (900)
D (936)
Explanation opens after your attempt
Step 1
Concept
The first term is (1716) and the second is (924). The difference is (792).
Step 2
Why this answer is correct
The correct answer is A. (792). The first term is (1716) and the second is (924). The difference is (792).
Step 3
Exam Tip
पहला पद (1716) और दूसरा (924) है। अंतर (792) है।
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( \frac{(n+4)!}{(n+1)!}+\frac{(n+3)!}{n!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+4)!}{(n+1)!}+\frac{(n+3)!}{n!} )?
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A ( (n+3)(n+2)(2n+4) )
B ( (n+4)(n+3)(2n+5) )
C ( (n+3)(n+2)(2n+5) )
D ( (n+2)(n+1)(2n+5) )
Explanation opens after your attempt
Correct Answer
C. ( (n+3)(n+2)(2n+5) )
Step 1
Concept
The common factor in both terms is ( (n+3)(n+2) ). Therefore the simplified form is ( (n+3)(n+2)(2n+5) ).
Step 2
Why this answer is correct
The correct answer is C. ( (n+3)(n+2)(2n+5) ). The common factor in both terms is ( (n+3)(n+2) ). Therefore the simplified form is ( (n+3)(n+2)(2n+5) ).
Step 3
Exam Tip
दोनों पदों में ( (n+3)(n+2) ) सामान्य है। इसलिए सरल रूप ( (n+3)(n+2)(2n+5) ) मिलता है।
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