Level 56 • 50/50 questions • 25 seconds per question.
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What is the value of \( \frac{19!}{16!}-\frac{18!}{15!} \)?
B. (918)
पहला पद (5814) और दूसरा पद (4896) है। इसलिए अंतर (918) है। / The first term is (5814) and the second term is (4896). Therefore the difference is (918).
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What is the simplified value of \( \frac{16!}{12!\cdot4!} \)?
B. (1820)
(16!) को \(16\cdot15\cdot14\cdot13\cdot12!\) लिखें। (4!) से भाग देने पर मान (1820) मिलता है। / Write (16!) as \(16\cdot15\cdot14\cdot13\cdot12!\). Dividing by (4!) gives (1820).
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If ( \frac{(n+5)!}{(n+1)!}=11880 ), what is the value of (n)?
B. (7)
यह ((n+5)(n+4)(n+3)(n+2)=11880) देता है। \(12\cdot11\cdot10\cdot9=11880\), इसलिए (n=7)। / It gives ((n+5)(n+4)(n+3)(n+2)=11880). Since \(12\cdot11\cdot10\cdot9=11880\), (n=7).
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If ( \frac{(n+4)!+(n+3)!}{(n+2)!}=143 ), what will be the value of (n)?
B. (8)
सरल रूप ((n+3)(n+5)) है। \(11\cdot13=143\), इसलिए (n=8)। / The simplified form is ((n+3)(n+5)). Since \(11\cdot13=143\), (n=8).
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What is the simplified form of ( \frac{(n+5)!-(n+4)!}{(n+3)!} )?
A. ( (n+4)2 )
((n+5)!-(n+4)!=(n+4)!((n+5)-1)) है। इसलिए भाग देने पर ((n+4)2) मिलता है। / ((n+5)!-(n+4)!=(n+4)!((n+5)-1)). Therefore division gives ((n+4)2).
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If ( \frac{(n+5)!-(n+4)!}{(n+3)!}=169 ), what is the value of (n)?
B. (9)
अभिव्यक्ति ((n+4)2) बनती है। (n+4=13), इसलिए (n=9)। / The expression becomes ((n+4)2). Since (n+4=13), (n=9).
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What is the simplified form of ( \frac{(n!)2}{(n-3)!(n+3)!} )?
B. ( \frac{n(n-1)(n-2)}{(n+1)(n+2)(n+3)} )
( \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। / ( \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.
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If ( \frac{(n!)2}{(n-3)!(n+3)!}=\frac{5}{21} ), what is the value of (n)?
B. (6)
सरल रूप में (n=6) रखने पर \( \frac{6\cdot5\cdot4}{7\cdot8\cdot9}=\frac{5}{21} \) मिलता है। विकल्प जांचते समय कटौती करें। / Putting (n=6) in the simplified form gives \( \frac{6\cdot5\cdot4}{7\cdot8\cdot9}=\frac{5}{21} \). Cancel while checking options.
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What is the highest power of (2) that divides (50!)?
C. (47)
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। / 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.
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How many zeros will be at the end of (60!)?
B. (14)
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। / 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).
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What is the smallest positive (n) for which (n!) is divisible by \(2^9\cdot3^3\cdot5^2\)?
C. (11)
(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। / (11!) has exponent (8) of (2), so \(2^9\) is not satisfied. Therefore the minimum is (12).
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What is the value of \( \frac{23!}{21!\cdot2!}+\frac{22!}{20!\cdot2!} \)?
B. (474)
दोनों पद (253) और (231) हैं। योग (484) नहीं बल्कि (484) है। / The two terms are (253) and (231). Their sum is (484).
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What is the value of \( \frac{24!}{21!\cdot3!}-\frac{23!}{20!\cdot3!} \)?
B. (252)
पहला पद (2024) और दूसरा (1771) नहीं, सही दूसरा (1771) है; अंतर (253) होगा। / The first term is (2024) and the second is (1771), so the difference is (253).
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What is the simplified value of \( \frac{15!}{11!\cdot4!}\div\frac{14!}{10!\cdot4!} \)?
A. \( \frac{15}{11} \)
दोनों पद (1365) और (1001) हैं। अनुपात \( \frac{1365}{1001}=\frac{15}{11} \) है। / The two terms are (1365) and (1001). The ratio is \( \frac{1365}{1001}=\frac{15}{11} \).
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How many factors remain in the expansion of ( \frac{(2n+3)!}{(2n-1)!} )?
B. (4)
विस्तार में ((2n+3)(2n+2)(2n+1)(2n)) बचता है। इसलिए कुल (4) गुणक हैं। / The expansion leaves ((2n+3)(2n+2)(2n+1)(2n)). Therefore there are (4) factors.
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If ( \frac{(2n+3)!}{(2n-1)!}=17160 ), what is the value of (n)?
B. (5)
यह ((2n+3)(2n+2)(2n+1)(2n)=17160) है। \(13\cdot12\cdot11\cdot10=17160\), इसलिए (n=5)। / It is ((2n+3)(2n+2)(2n+1)(2n)=17160). Since \(13\cdot12\cdot11\cdot10=17160\), (n=5).
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What is ( \frac{(4n+1)!}{(4n-1)!} ) equal to?
A. ( (4n+1)(4n) )
((4n+1)!=(4n+1)(4n)(4n-1)!) है। इसलिए दो गुणक बचते हैं। / ((4n+1)!=(4n+1)(4n)(4n-1)!). Therefore two factors remain.
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If ( \frac{(4n+1)!}{(4n-1)!}=420 ), what is the value of (n)?
B. (5)
यह ((4n+1)(4n)=420) है। \(21\cdot20=420\), इसलिए (4n=20) और (n=5)। / It is ((4n+1)(4n)=420). Since \(21\cdot20=420\), (4n=20) and (n=5).
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What is the simplified form of ( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!} )?
B. (3(n+5)(n+4))
सामान्य ((n+5)(n+4)) निकालने पर अंतर ((n+6)-(n+3)=3) है। इसलिए रूप (3(n+5)(n+4)) है। / Taking common ((n+5)(n+4)), the difference is ((n+6)-(n+3)=3). So the form is (3(n+5)(n+4)).
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If ( \frac{(n+6)!}{(n+3)!}-\frac{(n+5)!}{(n+2)!}=546 ), what is the value of (n)?
B. (8)
सरल रूप (3(n+5)(n+4)) है। \(3\cdot13\cdot12=468\) नहीं, सही \(3\cdot14\cdot13=546\) से (n=8) मिलता है। / The simplified form is (3(n+5)(n+4)). Since \(3\cdot13\cdot12=468\) is wrong and \(3\cdot14\cdot13=546\), (n=8).
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What is the simplified form of ( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!} )?
A. (2(n+5)2)
दोनों पद ((n+6)(n+5)) और ((n+5)(n+4)) हैं। समान ((n+5)) निकालने पर (2(n+5)2) मिलता है। / The two terms are ((n+6)(n+5)) and ((n+5)(n+4)). Taking common ((n+5)) gives (2(n+5)2).
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If ( \frac{(n+6)!}{(n+4)!}+\frac{(n+5)!}{(n+3)!}=338 ), what is the value of (n)?
A. (8)
सरल रूप (2(n+5)2) है। (2(n+5)2=338) से (n+5=13), इसलिए (n=8)। / The simplified form is (2(n+5)2). From (2(n+5)2=338), (n+5=13), so (n=8).
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What is the value of \( \frac{13!}{7!\cdot6!}+\frac{13!}{8!\cdot5!} \)?
B. (3432)
दोनों पद (1716) और (1287) नहीं, सही योग (3003) है। / The two terms are (1716) and (1287), so the correct sum is (3003).
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What is the value of \( \frac{12!}{6!\cdot6!}\times\frac{3}{11} \)?
C. (252)
पहला भाग (924) है। \(924\cdot\frac{3}{11}=252\) मिलता है। / The first part is (924). Then \(924\cdot\frac{3}{11}=252\).
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What is the value of \( \frac{14!}{11!}-3\cdot\frac{13!}{11!} \)?
B. (1716)
पहला पद \(14\cdot13\cdot12=2184\) और दूसरा \(3\cdot13\cdot12=468\) है। अंतर (1716) है। / The first term is \(14\cdot13\cdot12=2184\) and the second is \(3\cdot13\cdot12=468\). The difference is (1716).
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What is the value of \( \frac{10!+3\cdot9!}{8!} \)?
B. (117)
\( \frac{10!}{8!}=90 \) और \( \frac{3\cdot9!}{8!}=27 \) है। कुल (117) है। / \( \frac{10!}{8!}=90 \) and \( \frac{3\cdot9!}{8!}=27 \). The total is (117).
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What is the simplified form of ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)} )?
B. ( (n+1)(n+4) )
ऊपर ((n+4)(n+3)(n+2)(n+1)n(n-1)!) है। काटने पर (n(n+1)(n+4)) बचेगा। / The numerator is ((n+4)(n+3)(n+2)(n+1)n(n-1)!). After cancellation (n(n+1)(n+4)) remains.
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If ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=990 ), what is the value of (n)?
B. (9)
सरल रूप (n(n+1)(n+4)) है। \(9\cdot10\cdot13=1170\) नहीं, इसलिए सही विकल्पों में मान नहीं है। / The simplified form is (n(n+1)(n+4)). Since \(9\cdot10\cdot13=1170\), no given option satisfies the equation.
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If ( \frac{(n+4)!}{(n-1)!(n+3)(n+2)}=840 ), what is the value of (n)?
B. (8)
सरल रूप (n(n+1)(n+4)) है। \(8\cdot9\cdot12=864\) नहीं, इसलिए समीकरण त्रुटिपूर्ण है। / The simplified form is (n(n+1)(n+4)). Since \(8\cdot9\cdot12=864\), the equation is invalid.
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What is the highest power of (3) that divides (35!)?
B. (15)
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। / 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.
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What is the smallest positive (n) for which (n!) is divisible by \(2^{10}\cdot3^4\cdot5\cdot7\)?
C. (12)
(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। / (12!) has exponent (10) of (2) and exponent (5) of (3). Therefore all given conditions are first satisfied by (12!).
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What is the value of \( \frac{27!}{25!\cdot2!}-\frac{26!}{24!\cdot2!} \)?
C. (26)
पहला पद (351) और दूसरा (325) है। अंतर (26) है। / The first term is (351) and the second is (325). The difference is (26).
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What is the value of \( \frac{17!}{13!\cdot4!}-\frac{16!}{13!\cdot3!} \)?
A. (1540)
पहला पद (2380) और दूसरा (840) है। अंतर (1540) है। / The first term is (2380) and the second is (840). The difference is (1540).
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How many consecutive factors will be in the expansion of ( \frac{(n+2)!}{(n-4)!} )?
C. (6)
विस्तार ((n+2)(n+1)n(n-1)(n-2)(n-3)) है। इसलिए (6) लगातार गुणक मिलते हैं। / The expansion is ((n+2)(n+1)n(n-1)(n-2)(n-3)). Therefore there are (6) consecutive factors.
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If ( \frac{(n+2)!}{(n-2)!}=5040 ), what is the value of (n)?
B. (8)
यह ((n+2)(n+1)n(n-1)=5040) है। \(10\cdot9\cdot8\cdot7=5040\), इसलिए (n=8)। / It is ((n+2)(n+1)n(n-1)=5040). Since \(10\cdot9\cdot8\cdot7=5040\), (n=8).
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What is the simplified form of ( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}} )?
A. \( \frac{n+3}{n-2} \)
दोनों बड़े अनुपातों के समान गुणक कट जाते हैं। अंत में \( \frac{n+3}{n-2} \) बचता है। / The common factors of both large ratios cancel out. Finally \( \frac{n+3}{n-2} \) remains.
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If ( \frac{\frac{(n+3)!}{(n-2)!}}{\frac{(n+2)!}{(n-3)!}}=\frac{5}{2} ), what is the value of (n)?
B. (4)
सरल रूप \( \frac{n+3}{n-2} \) है। \( \frac{n+3}{n-2}=\frac{5}{2} \) से (n=4) मिलता है। / The simplified form is \( \frac{n+3}{n-2} \). From \( \frac{n+3}{n-2}=\frac{5}{2} \), we get (n=4).
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How many zeros will be at the end of (80!)?
B. (19)
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। / 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).
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What is the simplified value of \( \frac{24!}{22!}\div\frac{12!}{10!} \)?
B. \( \frac{46}{11} \)
मान \( \frac{24\cdot23}{12\cdot11}=\frac{46}{11} \) है। भाग में दोनों फैक्टोरियल अनुपात फैलाएं। / The value is \( \frac{24\cdot23}{12\cdot11}=\frac{46}{11} \). Expand both factorial ratios in division.
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What is the simplified value of \( \frac{8!\cdot11!}{10!\cdot7!} \)?
C. (88)
\( \frac{8!}{7!}=8 \) और \( \frac{11!}{10!}=11 \) है। इसलिए मान (88) है। / \( \frac{8!}{7!}=8 \) and \( \frac{11!}{10!}=11 \). Therefore the value is (88).
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What is the simplified form of ( \frac{(n+6)!-(n+5)!}{(n+4)!} )?
A. ( (n+5)2 )
((n+6)!-(n+5)!=(n+5)!((n+6)-1)) है। इसलिए उत्तर ((n+5)2) है। / ((n+6)!-(n+5)!=(n+5)!((n+6)-1)). Therefore the answer is ((n+5)2).
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If ( \frac{(n+6)!-(n+5)!}{(n+4)!}=196 ), what is the value of (n)?
B. (9)
सरल रूप ((n+5)2) है। (n+5=14), इसलिए (n=9)। / The simplified form is ((n+5)2). Since (n+5=14), (n=9).
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What is the simplified form of ( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!} )?
B. (4(n+6)(n+5))
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। / Take common ((n+6)(n+5)). The difference is ((n+7)-(n+4)=3), so the correct form is (3(n+6)(n+5)).
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If ( \frac{(n+7)!}{(n+4)!}-\frac{(n+6)!}{(n+3)!}=720 ), what is the value of (n)?
A. (9)
सरल रूप (3(n+6)(n+5)) है। \(3\cdot15\cdot14=630\) नहीं, इसलिए सही विकल्प नहीं है। / The simplified form is (3(n+6)(n+5)). Since \(3\cdot15\cdot14=630\), no option is correct.
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If ( \frac{(n!)2}{(n-1)!(n+1)!}=\frac{13}{14} ), what is the value of (n)?
B. (13)
सरल रूप \( \frac{n}{n+1} \) है। \( \frac{n}{n+1}=\frac{13}{14} \), इसलिए (n=13)। / The simplified form is \( \frac{n}{n+1} \). Since \( \frac{n}{n+1}=\frac{13}{14} \), (n=13).
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What is the simplified value of \( \frac{19!}{15!\cdot4!}\div\frac{18!}{14!\cdot4!} \)?
B. \( \frac{19}{15} \)
दोनों पदों का अनुपात सीधे \( \frac{19}{15} \) बनता है। बड़े मान निकालने के बजाय फैक्टोरियल काटें। / The ratio of the two terms directly becomes \( \frac{19}{15} \). Cancel factorials instead of finding large values.
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What is the simplified form of ( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!} )?
A. \(2n^3+6n^2+4n+6\)
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। / The first term is ((n+3)(n+2)(n+1)) and the second is ((n+1)n(n-1)). Adding gives \(2n^3+6n^2+4n+6\).
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If ( \frac{(n+3)!}{n!}+\frac{(n+1)!}{(n-2)!}=546 ), what is the value of (n)?
A. (6)
(n=6) रखने पर \(9\cdot8\cdot7+7\cdot6\cdot5=504+210=714\) नहीं मिलता। इसलिए यह समीकरण विकल्पों से मेल नहीं खाता। / Putting (n=6) gives \(9\cdot8\cdot7+7\cdot6\cdot5=504+210=714\). Therefore the equation does not match the options.
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What is the value of \( \frac{13!}{6!\cdot7!}-\frac{12!}{6!\cdot6!} \)?
A. (792)
पहला पद (1716) और दूसरा (924) है। अंतर (792) है। / The first term is (1716) and the second is (924). The difference is (792).
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What is the simplified form of ( \frac{(n+4)!}{(n+1)!}+\frac{(n+3)!}{n!} )?
C. ( (n+3)(n+2)(2n+5) )
दोनों पदों में ( (n+3)(n+2) ) सामान्य है। इसलिए सरल रूप ( (n+3)(n+2)(2n+5) ) मिलता है। / The common factor in both terms is ( (n+3)(n+2) ). Therefore the simplified form is ( (n+3)(n+2)(2n+5) ).
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