Level 55 • 50/50 questions • 25 seconds per question.
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What is the value of \( \frac{18!}{15!}-\frac{17!}{14!} \)?
B. (816)
पहला पद \(18\cdot17\cdot16=4896\) और दूसरा \(17\cdot16\cdot15=4080\) है। अंतर (816) है। / The first term is \(18\cdot17\cdot16=4896\) and the second is \(17\cdot16\cdot15=4080\). The difference is (816).
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What is the simplified value of \( \frac{15!}{11!\cdot4!} \)?
C. (1365)
\( \frac{15\cdot14\cdot13\cdot12}{4!}=1365 \) है। हर में (4!) का सही मान लगाएं। / \( \frac{15\cdot14\cdot13\cdot12}{4!}=1365 \). Use the correct value of (4!) in the denominator.
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If ( \frac{(n+4)!}{n!}=11880 ), what will be the value of (n)?
C. (8)
यह ((n+4)(n+3)(n+2)(n+1)=11880) है और \(12\cdot11\cdot10\cdot9=11880\)। इसलिए (n=8) है। / It is ((n+4)(n+3)(n+2)(n+1)=11880), and \(12\cdot11\cdot10\cdot9=11880\). So (n=8).
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If ( \frac{(n+2)!+(n+1)!}{(n+1)!}=14 ), what is the value of (n)?
C. (11)
सरल रूप (n+3) है। इसलिए (n+3=14) और (n=11)। / The simplified form is (n+3). Thus (n+3=14) and (n=11).
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What is the simplified form of ( \frac{(n+3)!-(n+2)!}{(n+1)!} )?
A. ((n+2)2)
((n+3)!-(n+2)!=(n+2)!((n+3)-1)) है। इससे ((n+2)2) मिलता है। / ((n+3)!-(n+2)!=(n+2)!((n+3)-1)). This gives ((n+2)2).
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If ( \frac{(n+3)!-(n+2)!}{(n+1)!}=121 ), what will be the value of (n)?
B. (9)
सरल रूप ((n+2)2) है। इसलिए (n+2=11) और (n=9)। / The simplified form is ((n+2)2). Therefore (n+2=11) and (n=9).
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If ( \frac{(n!)2}{(n-2)!(n+2)!}=\frac{10}{21} ), what is the value of (n)?
B. (5)
सरल रूप ( \frac{n(n-1)}{(n+1)(n+2)} ) है। (n=5) रखने पर \( \frac{20}{42}=\frac{10}{21} \) मिलता है। / The simplified form is ( \frac{n(n-1)}{(n+1)(n+2)} ). Putting (n=5) gives \( \frac{20}{42}=\frac{10}{21} \).
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What is the highest power of (2) that divides (20!)?
D. (18)
घात \( \left\lfloor\frac{20}{2}\right\rfloor+\left\lfloor\frac{20}{4}\right\rfloor+\left\lfloor\frac{20}{8}\right\rfloor+\left\lfloor\frac{20}{16}\right\rfloor=18 \) है। अभाज्य घात के लिए सभी भागफल जोड़ें। / The exponent is \( \left\lfloor\frac{20}{2}\right\rfloor+\left\lfloor\frac{20}{4}\right\rfloor+\left\lfloor\frac{20}{8}\right\rfloor+\left\lfloor\frac{20}{16}\right\rfloor=18 \). Add all quotients for a prime exponent.
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How many zeros will be at the end of (45!)?
A. (10)
शून्यों की संख्या \( \left\lfloor\frac{45}{5}\right\rfloor+\left\lfloor\frac{45}{25}\right\rfloor=10 \) है। (25) के योगदान को न भूलें। / The number of zeros is \( \left\lfloor\frac{45}{5}\right\rfloor+\left\lfloor\frac{45}{25}\right\rfloor=10 \). Do not forget the contribution of (25).
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What is the smallest positive (n) for which (n!) is divisible by (1440)?
C. (8)
\(1440=2^5\cdot3^2\cdot5\) है और यह जरूरत पहली बार (8!) में पूरी होती है। विभाज्यता में अभाज्य गुणनखंड जांचें। / \(1440=2^5\cdot3^2\cdot5\), and this requirement is first satisfied by (8!). For divisibility, check prime factors.
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What is the value of \( \frac{20!}{18!\cdot2!}+\frac{19!}{17!\cdot2!} \)?
C. (361)
दोनों पद (190) और (171) हैं। योग (361) है। / The two terms are (190) and (171). Their sum is (361).
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What is the value of \( \frac{21!}{18!\cdot3!}-\frac{20!}{17!\cdot3!} \)?
C. (190)
पहला पद (1330) और दूसरा (1140) है। अंतर (190) है। / The first term is (1330) and the second is (1140). The difference is (190).
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What is the simplified value of \( \frac{14!}{10!\cdot4!}\div\frac{13!}{9!\cdot4!} \)?
B. \(\frac{7}{5}\)
दोनों पद (1001) और (715) हैं। अनुपात \( \frac{1001}{715}=\frac{7}{5} \) है। / The two terms are (1001) and (715). The ratio is \( \frac{1001}{715}=\frac{7}{5} \).
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Which is the correct expansion of ( \frac{(2n)!}{(2n-4)!} )?
B. ((2n)(2n-1)(2n-2)(2n-3))
((2n)!) को ((2n)(2n-1)(2n-2)(2n-3)(2n-4)!) तक फैलाते हैं। इसलिए चार गुणक बचते हैं। / We expand ((2n)!) up to ((2n)(2n-1)(2n-2)(2n-3)(2n-4)!). Therefore four factors remain.
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If ( \frac{(2n)!}{(2n-4)!}=1680 ), what is the value of (n)?
B. (4)
यह ((2n)(2n-1)(2n-2)(2n-3)=1680) है। \(8\cdot7\cdot6\cdot5=1680\), इसलिए (n=4)। / It is ((2n)(2n-1)(2n-2)(2n-3)=1680). Since \(8\cdot7\cdot6\cdot5=1680\), (n=4).
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What is ( \frac{(3n+2)!}{(3n)!} ) equal to?
A. ((3n+2)(3n+1))
((3n+2)!=(3n+2)(3n+1)(3n)!) है। इसलिए दो गुणक बचते हैं। / ((3n+2)!=(3n+2)(3n+1)(3n)!). Therefore two factors remain.
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If ( \frac{(3n+2)!}{(3n)!}=182 ), what is the value of (n)?
B. (4)
यह ((3n+2)(3n+1)=182) है और \(14\cdot13=182\)। इसलिए (3n+2=14) और (n=4)। / It is ((3n+2)(3n+1)=182), and \(14\cdot13=182\). Thus (3n+2=14) and (n=4).
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If ( \frac{(n+4)!}{(n+2)!}-\frac{(n+3)!}{(n+1)!}=24 ), what is the value of (n)?
C. (9)
सरल रूप (2(n+3)) है। इसलिए (2(n+3)=24) और (n=9)। / The simplified form is (2(n+3)). Thus (2(n+3)=24) and (n=9).
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What is the simplified form of ( \frac{(n+5)!+(n+4)!}{(n+3)!} )?
A. ((n+4)(n+6))
ऊपर ((n+4)!((n+5)+1)) बनता है। भाग देने पर ((n+4)(n+6)) मिलता है। / The numerator becomes ((n+4)!((n+5)+1)). Dividing gives ((n+4)(n+6)).
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If ( \frac{(n+5)!+(n+4)!}{(n+3)!}=120 ), what is the value of (n)?
B. (6)
सरल रूप ((n+4)(n+6)) है। (n=6) रखने पर \(10\cdot12=120\) मिलता है। / The simplified form is ((n+4)(n+6)). Putting (n=6) gives \(10\cdot12=120\).
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What is the value of \( \frac{12!}{8!\cdot4!}+\frac{12!}{9!\cdot3!} \)?
B. (715)
दोनों पद (495) और (220) हैं। योग (715) है। / The two terms are (495) and (220). Their sum is (715).
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What is the value of \( \frac{10!}{5!\cdot5!}\times\frac{5}{9} \)?
C. (140)
पहला भाग (252) है। \(252\cdot\frac{5}{9}=140\) मिलता है। / The first part is (252). Then \(252\cdot\frac{5}{9}=140\).
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What is the value of \( \frac{13!}{10!}-2\cdot\frac{12!}{10!} \)?
B. (1452)
पहला पद (1716) और दूसरा \(2\cdot132=264\) है। अंतर (1452) है। / The first term is (1716) and the second is \(2\cdot132=264\). The difference is (1452).
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What is the value of \( \frac{9!+2\cdot8!}{7!} \)?
C. (88)
\( \frac{9!}{7!}=72 \) और \( \frac{2\cdot8!}{7!}=16 \) है। कुल (88) मिलता है। / \( \frac{9!}{7!}=72 \) and \( \frac{2\cdot8!}{7!}=16 \). The total is (88).
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What is the simplified form of ( \frac{(n+3)!}{(n-1)!(n+2)(n+1)} )?
A. (n(n+3))
ऊपर ((n+3)(n+2)(n+1)n(n-1)!) है। काटने पर (n(n+3)) बचता है। / The numerator is ((n+3)(n+2)(n+1)n(n-1)!). After cancellation, (n(n+3)) remains.
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If ( \frac{(n+3)!}{(n-1)!(n+2)(n+1)}=108 ), what is the value of (n)?
C. (9)
सरल रूप (n(n+3)) है। \(9\cdot12=108\), इसलिए (n=9)। / The simplified form is (n(n+3)). Since \(9\cdot12=108\), (n=9).
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What is the highest power of (3) that divides (25!)?
C. (10)
घात \( \left\lfloor\frac{25}{3}\right\rfloor+\left\lfloor\frac{25}{9}\right\rfloor=10 \) है। (9) के योगदान को जोड़ना जरूरी है। / The exponent is \( \left\lfloor\frac{25}{3}\right\rfloor+\left\lfloor\frac{25}{9}\right\rfloor=10 \). Adding the contribution of (9) is necessary.
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What is the smallest positive (n) for which (n!) is divisible by \(2^7\cdot3^4\cdot5\)?
B. (9)
(9!) में (2) की घात (7) और (3) की घात (4) मिल जाती है। इससे छोटा (n) यह शर्त पूरी नहीं करता। / (9!) contains exponent (7) of (2) and exponent (4) of (3). No smaller (n) satisfies this condition.
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What is the value of \( \frac{25!}{23!\cdot2!}-\frac{24!}{22!\cdot2!} \)?
C. (24)
पहला पद (300) और दूसरा (276) है। अंतर (24) है। / The first term is (300) and the second is (276). The difference is (24).
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What is the value of \( \frac{16!}{12!\cdot4!}-\frac{15!}{12!\cdot3!} \)?
B. (1365)
पहला पद (1820) और दूसरा (455) है। अंतर (1365) है। / The first term is (1820) and the second is (455). The difference is (1365).
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Which is the correct expansion of ( \frac{(n+1)!}{(n-3)!} )?
A. ((n+1)n(n-1)(n-2))
((n+1)!) को ((n+1)n(n-1)(n-2)(n-3)!) तक फैलाते हैं। इसलिए चार गुणक बचते हैं। / We expand ((n+1)!) up to ((n+1)n(n-1)(n-2)(n-3)!). Therefore four factors remain.
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If ( \frac{(n+1)!}{(n-3)!}=5040 ), what is the value of (n)?
C. (9)
यह ((n+1)n(n-1)(n-2)=5040) है। \(10\cdot9\cdot8\cdot7=5040\), इसलिए (n=9)। / It is ((n+1)n(n-1)(n-2)=5040). Since \(10\cdot9\cdot8\cdot7=5040\), (n=9).
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What is the simplified form of ( \frac{\frac{(n+2)!}{(n-2)!}}{\frac{(n+1)!}{(n-3)!}} )?
A. \(\frac{n+2}{n-2}\)
समान गुणक ((n+1)n(n-1)) कट जाते हैं। इसलिए \( \frac{n+2}{n-2} \) बचता है। / The common factors ((n+1)n(n-1)) cancel out. Therefore \( \frac{n+2}{n-2} \) remains.
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If ( \frac{\frac{(n+2)!}{(n-2)!}}{\frac{(n+1)!}{(n-3)!}}=2 ), what is the value of (n)?
C. (6)
सरल रूप \( \frac{n+2}{n-2} \) है। \( \frac{n+2}{n-2}=2 \) से (n=6) मिलता है। / The simplified form is \( \frac{n+2}{n-2} \). From \( \frac{n+2}{n-2}=2 \), we get (n=6).
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How many zeros will be at the end of (30!)?
B. (7)
शून्यों की संख्या \( \left\lfloor\frac{30}{5}\right\rfloor+\left\lfloor\frac{30}{25}\right\rfloor=7 \) है। (25) का अतिरिक्त योगदान जोड़ें। / The number of zeros is \( \left\lfloor\frac{30}{5}\right\rfloor+\left\lfloor\frac{30}{25}\right\rfloor=7 \). Add the extra contribution of (25).
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What is the simplified value of \( \frac{22!}{20!}\div\frac{11!}{9!} \)?
B. \(\frac{21}{5}\)
मान \( \frac{22\cdot21}{11\cdot10}=\frac{21}{5} \) है। भाग में दोनों अनुपातों को फैलाकर काटें। / The value is \( \frac{22\cdot21}{11\cdot10}=\frac{21}{5} \). Expand and cancel both ratios in division.
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What is the simplified value of \( \frac{7!\cdot9!}{8!\cdot6!} \)?
B. (63)
\( \frac{7!}{6!}=7 \) और \( \frac{9!}{8!}=9 \) है। इसलिए मान (63) है। / \( \frac{7!}{6!}=7 \) and \( \frac{9!}{8!}=9 \). Hence the value is (63).
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What is the simplified form of ( \frac{(n+4)!-(n+3)!}{(n+2)!} )?
A. ((n+3)2)
((n+4)!-(n+3)!=(n+3)!((n+4)-1)) है। इसलिए ((n+3)2) मिलता है। / ((n+4)!-(n+3)!=(n+3)!((n+4)-1)). Therefore ((n+3)2) is obtained.
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If ( \frac{(n+4)!-(n+3)!}{(n+2)!}=144 ), what is the value of (n)?
B. (9)
सरल रूप ((n+3)2) है। इसलिए (n+3=12) और (n=9)। / The simplified form is ((n+3)2). Thus (n+3=12) and (n=9).
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What is the simplified form of ( \frac{(n+5)!}{(n+2)!}-\frac{(n+4)!}{(n+1)!} )?
B. (3(n+4)(n+3))
सामान्य ((n+4)(n+3)) निकालने पर अंतर ((n+5)-(n+2)=3) है। उत्तर (3(n+4)(n+3)) है। / Taking common ((n+4)(n+3)), the difference is ((n+5)-(n+2)=3). The answer is (3(n+4)(n+3)).
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If ( \frac{(n+5)!}{(n+2)!}-\frac{(n+4)!}{(n+1)!}=396 ), what is the value of (n)?
B. (8)
सरल रूप (3(n+4)(n+3)) है। \(3\cdot12\cdot11=396\), इसलिए (n=8)। / The simplified form is (3(n+4)(n+3)). Since \(3\cdot12\cdot11=396\), (n=8).
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If ( \frac{(n!)2}{(n-1)!(n+1)!}=\frac{11}{12} ), what is the value of (n)?
B. (11)
सरल रूप \( \frac{n}{n+1} \) है। \( \frac{n}{n+1}=\frac{11}{12} \) से (n=11) मिलता है। / The simplified form is \( \frac{n}{n+1} \). From \( \frac{n}{n+1}=\frac{11}{12} \), we get (n=11).
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What is the simplified value of \( \frac{18!}{14!\cdot4!}\div\frac{17!}{13!\cdot4!} \)?
A. \(\frac{153}{119}\)
दोनों पद (3060) और (2380) हैं। उनका अनुपात \( \frac{3060}{2380}=\frac{153}{119} \) है। / The two terms are (3060) and (2380). Their ratio is \( \frac{3060}{2380}=\frac{153}{119} \).
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What is the simplified form of ( \frac{(n+2)!}{n!}+\frac{n!}{(n-2)!} )?
B. \(2n^2+2n+2\)
पहला पद ((n+2)(n+1)) और दूसरा (n(n-1)) है। जोड़ने पर \(2n^2+2n+2\) मिलता है। / The first term is ((n+2)(n+1)) and the second is (n(n-1)). Adding gives \(2n^2+2n+2\).
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If ( \frac{(n+2)!}{n!}+\frac{n!}{(n-2)!}=114 ), what is the value of (n)?
B. (7)
सरल रूप \(2n^2+2n+2\) है। (n=7) रखने पर (98+14+2=114) मिलता है। / The simplified form is \(2n^2+2n+2\). Putting (n=7) gives (98+14+2=114).
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What is the value of \( \frac{12!}{6!\cdot6!}-\frac{11!}{5!\cdot6!} \)?
C. (462)
पहला पद (924) और दूसरा (462) है। अंतर (462) है। / The first term is (924) and the second is (462). The difference is (462).
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What is the simplified form of ( \frac{(n+6)!}{(n+2)!}-\frac{(n+5)!}{(n+1)!} )?
A. (4(n+5)(n+4)(n+3))
सामान्य ((n+5)(n+4)(n+3)) निकालने पर अंतर ((n+6)-(n+2)=4) है। पहले समान गुणकों को पहचानें। / Taking common ((n+5)(n+4)(n+3)), the difference is ((n+6)-(n+2)=4). Identify common factors first.
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If ( \frac{(n+6)!}{(n+2)!}-\frac{(n+5)!}{(n+1)!}=2880 ), what is the value of (n)?
B. (5)
सरल रूप (4(n+5)(n+4)(n+3)) है। \(4\cdot10\cdot9\cdot8=2880\), इसलिए (n=5)। / The simplified form is (4(n+5)(n+4)(n+3)). Since \(4\cdot10\cdot9\cdot8=2880\), (n=5).
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What is the smallest positive (n) for which (n!) is divisible by \(2^8\cdot3^2\cdot5^2\)?
C. (10)
(10!) में (2) की घात (8) और (5) की घात (2) मिलती है। (9!) में \(5^2\) नहीं आता, इसलिए (10) न्यूनतम है। / (10!) contains exponent (8) of (2) and exponent (2) of (5). (9!) does not contain \(5^2\), so (10) is minimum.
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What is the highest power of (7) that divides (100!)?
C. (16)
घात \( \left\lfloor\frac{100}{7}\right\rfloor+\left\lfloor\frac{100}{49}\right\rfloor=16 \) है। उच्च घातों जैसे (49) का योगदान जरूर जोड़ें। / The exponent is \( \left\lfloor\frac{100}{7}\right\rfloor+\left\lfloor\frac{100}{49}\right\rfloor=16 \). Always add the contribution of higher powers such as (49).
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