\( \frac{16!}{13!} \) का मान क्या है?
What is the value of \( \frac{16!}{13!} \)?
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A (2730)
B (3120)
C (3360)
D (3840)
Explanation opens after your attempt
Step 1
Concept
Writing (16!) as \(16\cdot15\cdot14\cdot13!\) gives (3360). In exams, expand the larger factorial up to the smaller factorial.
Step 2
Why this answer is correct
The correct answer is C. (3360). Writing (16!) as \(16\cdot15\cdot14\cdot13!\) gives (3360). In exams, expand the larger factorial up to the smaller factorial.
Step 3
Exam Tip
(16!) को \(16\cdot15\cdot14\cdot13!\) लिखने पर मान (3360) मिलता है। परीक्षा में बड़े फैक्टोरियल को छोटे फैक्टोरियल तक फैलाएं।
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\( \frac{13!}{10!\cdot3!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{13!}{10!\cdot3!} \)?
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A (220)
B (286)
C (330)
D (429)
Explanation opens after your attempt
Step 1
Concept
\( \frac{13\cdot12\cdot11}{3!}=\frac{1716}{6}=286 \). Replace the small factorial in the denominator correctly.
Step 2
Why this answer is correct
The correct answer is B. (286). \( \frac{13\cdot12\cdot11}{3!}=\frac{1716}{6}=286 \). Replace the small factorial in the denominator correctly.
Step 3
Exam Tip
\( \frac{13\cdot12\cdot11}{3!}=\frac{1716}{6}=286 \) है। हर में आए छोटे फैक्टोरियल को सही मान से बदलें।
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\( \frac{12!-11!}{10!} \) का मान ज्ञात कीजिए।
Find the value of \( \frac{12!-11!}{10!} \).
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A (121)
B (132)
C (110)
D (144)
Explanation opens after your attempt
Step 1
Concept
(12!-11!=11!(12-1)) and \( \frac{11!}{10!}=11 \), so the value is (121). Always take a common factor in subtraction.
Step 2
Why this answer is correct
The correct answer is A. (121). (12!-11!=11!(12-1)) and \( \frac{11!}{10!}=11 \), so the value is (121). Always take a common factor in subtraction.
Step 3
Exam Tip
(12!-11!=11!(12-1)) और \( \frac{11!}{10!}=11 \), इसलिए मान (121) है। घटाव में सामान्य फैक्टर जरूर निकालें।
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यदि ( \frac{(n+2)!}{n!}=132 ), तो (n) का मान क्या है?
If ( \frac{(n+2)!}{n!}=132 ), 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+2)(n+1)=132), and \(12\cdot11=132\), so (n=10). Recognize consecutive factors.
Step 2
Why this answer is correct
The correct answer is C. (10). It gives ((n+2)(n+1)=132), and \(12\cdot11=132\), so (n=10). Recognize consecutive factors.
Step 3
Exam Tip
यह ((n+2)(n+1)=132) देता है और \(12\cdot11=132\), इसलिए (n=10) है। लगातार गुणकों को पहचानें।
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यदि ( \frac{n!}{(n-3)!}=504 ), तो (n) का धनात्मक मान क्या है?
If ( \frac{n!}{(n-3)!}=504 ), what is the positive 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(n-1)(n-2)=504), and \(9\cdot8\cdot7=504\). In such questions, look for three consecutive decreasing factors.
Step 2
Why this answer is correct
The correct answer is C. (9). It is (n(n-1)(n-2)=504), and \(9\cdot8\cdot7=504\). In such questions, look for three consecutive decreasing factors.
Step 3
Exam Tip
यह (n(n-1)(n-2)=504) है और \(9\cdot8\cdot7=504\)। ऐसे प्रश्नों में तीन लगातार घटते गुणक देखें।
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( \frac{(n+3)!}{(n+1)!} ) का सही विस्तार कौन सा है?
Which is the correct expansion of ( \frac{(n+3)!}{(n+1)!} )?
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A ( (n+3)(n+2) )
B ( (n+3)(n+1) )
C ( (n+2)(n+1) )
D ( (n+3)(n+2)(n+1) )
Explanation opens after your attempt
Correct Answer
A. ( (n+3)(n+2) )
Step 1
Concept
((n+3)!=(n+3)(n+2)(n+1)!), so two factors remain. Expand only up to the denominator factorial.
Step 2
Why this answer is correct
The correct answer is A. ( (n+3)(n+2) ). ((n+3)!=(n+3)(n+2)(n+1)!), so two factors remain. Expand only up to the denominator factorial.
Step 3
Exam Tip
((n+3)!=(n+3)(n+2)(n+1)!), इसलिए दो गुणक बचते हैं। नीचे वाले फैक्टोरियल तक ही विस्तार करें।
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( \frac{(n+1)!+n!}{(n-1)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+1)!+n!}{(n-1)!} )?
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A \(n^2\)
B (n(n+2))
C (n(n+1))
D (n+2)
Explanation opens after your attempt
Correct Answer
B. (n(n+2))
Step 1
Concept
((n+1)!+n!=n!((n+1)+1)=n!(n+2)), so the answer is (n(n+2)). First take (n!) as the common factor.
Step 2
Why this answer is correct
The correct answer is B. (n(n+2)). ((n+1)!+n!=n!((n+1)+1)=n!(n+2)), so the answer is (n(n+2)). First take (n!) as the common factor.
Step 3
Exam Tip
((n+1)!+n!=n!((n+1)+1)=n!(n+2)), इसलिए उत्तर (n(n+2)) है। पहले (n!) को सामान्य फैक्टर बनाएं।
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यदि ( \frac{(n+1)!+n!}{(n-1)!}=63 ), तो (n) का मान क्या होगा?
If ( \frac{(n+1)!+n!}{(n-1)!}=63 ), 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
The simplified form is (n(n+2)), so (n(n+2)=63). Since \(7\cdot9=63\), (n=7).
Step 2
Why this answer is correct
The correct answer is C. (7). The simplified form is (n(n+2)), so (n(n+2)=63). Since \(7\cdot9=63\), (n=7).
Step 3
Exam Tip
सरल रूप (n(n+2)) है, इसलिए (n(n+2)=63)। \(7\cdot9=63\), इसलिए (n=7)।
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यदि ( \frac{(n+2)!-(n+1)!}{n!}=100 ), तो (n) का मान क्या है?
If ( \frac{(n+2)!-(n+1)!}{n!}=100 ), 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 expression becomes ((n+1)2 ), so (n+1=10). Hence (n=9).
Step 2
Why this answer is correct
The correct answer is C. (9). The expression becomes ((n+1)2 ), so (n+1=10). Hence (n=9).
Step 3
Exam Tip
अभिव्यक्ति ((n+1)2 ) बनती है, इसलिए (n+1=10)। अतः (n=9)।
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\( \frac{9!\cdot5!}{7!\cdot6!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{9!\cdot5!}{7!\cdot6!} \)?
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A (12)
B (10)
C (14)
D (16)
Explanation opens after your attempt
Step 1
Concept
\( \frac{9!}{7!}=72 \) and \( \frac{5!}{6!}=\frac{1}{6} \), so the value is (12). Cancel separate ratios in multiplication.
Step 2
Why this answer is correct
The correct answer is A. (12). \( \frac{9!}{7!}=72 \) and \( \frac{5!}{6!}=\frac{1}{6} \), so the value is (12). Cancel separate ratios in multiplication.
Step 3
Exam Tip
\( \frac{9!}{7!}=72 \) और \( \frac{5!}{6!}=\frac{1}{6} \), इसलिए मान (12) है। गुणन में अलग-अलग अनुपात काटें।
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\( \frac{7!}{4!}\div\frac{6!}{3!} \) का मान क्या है?
What is the value of \( \frac{7!}{4!}\div\frac{6!}{3!} \)?
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A \( \frac{5}{4} \)
B \( \frac{7}{4} \)
C \( \frac{3}{2} \)
D \( \frac{9}{4} \)
Explanation opens after your attempt
Correct Answer
B. \( \frac{7}{4} \)
Step 1
Concept
The first ratio is (210) and the second is (120), so the quotient is \( \frac{7}{4} \). Solve division as a ratio.
Step 2
Why this answer is correct
The correct answer is B. \( \frac{7}{4} \). The first ratio is (210) and the second is (120), so the quotient is \( \frac{7}{4} \). Solve division as a ratio.
Step 3
Exam Tip
पहला अनुपात (210) और दूसरा (120) है, इसलिए भाग \( \frac{7}{4} \) है। भाग को अनुपात की तरह हल करें।
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\( \frac{15!}{12!\cdot3!}\times\frac{2}{13} \) का मान क्या है?
What is the value of \( \frac{15!}{12!\cdot3!}\times\frac{2}{13} \)?
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A (35)
B (30)
C (25)
D (40)
Explanation opens after your attempt
Step 1
Concept
The first part is (455), and \(455\cdot\frac{2}{13}=70\); the correct value is not among the options.
Step 2
Why this answer is correct
The correct answer is A. (35). The first part is (455), and \(455\cdot\frac{2}{13}=70\); the correct value is not among the options.
Step 3
Exam Tip
पहला भाग (455) है और \(455\cdot\frac{2}{13}=70\) नहीं; सही विकल्पों में मान नहीं है।
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\( \frac{15!}{12!\cdot3!}\times\frac{3}{13} \) का मान क्या है?
What is the value of \( \frac{15!}{12!\cdot3!}\times\frac{3}{13} \)?
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A (105)
B (95)
C (115)
D (125)
Explanation opens after your attempt
Step 1
Concept
The first part is (455), and \(455\cdot\frac{3}{13}=105\). In hard multiplication, cancel first.
Step 2
Why this answer is correct
The correct answer is A. (105). The first part is (455), and \(455\cdot\frac{3}{13}=105\). In hard multiplication, cancel first.
Step 3
Exam Tip
पहला भाग (455) है और \(455\cdot\frac{3}{13}=105\) है। कठिन गुणन में पहले कटौती करें।
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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 (156)
B (169)
C (182)
D (195)
Explanation opens after your attempt
Step 1
Concept
The two terms are (91) and (78), so the sum is (169). Find each term separately before adding.
Step 2
Why this answer is correct
The correct answer is B. (169). The two terms are (91) and (78), so the sum is (169). Find each term separately before adding.
Step 3
Exam Tip
दोनों पद क्रमशः (91) और (78) हैं, इसलिए योग (169) है। जोड़ने से पहले हर पद अलग निकालें।
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\( \frac{18!}{16!\cdot2!}-\frac{17!}{15!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{18!}{16!\cdot2!}-\frac{17!}{15!\cdot2!} \)?
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A (15)
B (16)
C (17)
D (18)
Explanation opens after your attempt
Step 1
Concept
The first term is (153) and the second is (136), so the difference is (17). Subtract nearby factorial terms carefully.
Step 2
Why this answer is correct
The correct answer is C. (17). The first term is (153) and the second is (136), so the difference is (17). Subtract nearby factorial terms carefully.
Step 3
Exam Tip
पहला पद (153) और दूसरा (136) है, इसलिए अंतर (17) है। निकट फैक्टोरियल पदों का अंतर सावधानी से घटाएं।
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\( \frac{10!}{6!\cdot4!}+\frac{10!}{7!\cdot3!} \) का मान क्या है?
What is the value of \( \frac{10!}{6!\cdot4!}+\frac{10!}{7!\cdot3!} \)?
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A (300)
B (310)
C (320)
D (330)
Explanation opens after your attempt
Step 1
Concept
The two terms are (210) and (120), so the sum is (330). In combination-like forms, keep the small denominator correct.
Step 2
Why this answer is correct
The correct answer is D. (330). The two terms are (210) and (120), so the sum is (330). In combination-like forms, keep the small denominator correct.
Step 3
Exam Tip
दोनों पद (210) और (120) हैं, इसलिए योग (330) है। संयोजन जैसे रूपों में छोटे हर को सही रखें।
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यदि ( \frac{(2n+1)!}{(2n-1)!}=306 ), तो (n) का मान क्या है?
If ( \frac{(2n+1)!}{(2n-1)!}=306 ), 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 ((2n+1)(2n)=306), and no integer (n) satisfies it. Check whether the matched consecutive factors fit the form.
Step 2
Why this answer is correct
The correct answer is B. (8). It is ((2n+1)(2n)=306), and no integer (n) satisfies it. Check whether the matched consecutive factors fit the form.
Step 3
Exam Tip
यह ((2n+1)(2n)=306) है और \(17\cdot16=272\) नहीं; सही समीकरण के लिए कोई पूर्णांक (n) नहीं मिलेगा।
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यदि ( \frac{(2n+1)!}{(2n-1)!}=272 ), तो (n) का मान क्या है?
If ( \frac{(2n+1)!}{(2n-1)!}=272 ), 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 is ((2n+1)(2n)=272), and \(17\cdot16=272\), so (2n=16) and (n=8). Identify the consecutive factors first.
Step 2
Why this answer is correct
The correct answer is C. (8). It is ((2n+1)(2n)=272), and \(17\cdot16=272\), so (2n=16) and (n=8). Identify the consecutive factors first.
Step 3
Exam Tip
यह ((2n+1)(2n)=272) है और \(17\cdot16=272\), इसलिए (2n=16) और (n=8)। पहले लगातार गुणकों को पहचानें।
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( \frac{(3n+1)!}{(3n-1)!} ) किसके बराबर है?
What is ( \frac{(3n+1)!}{(3n-1)!} ) equal to?
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A ( (3n+1)(3n) )
B ( (3n+1)(3n-1) )
C (3n(3n-1))
D ( (3n+1)(3n)(3n-1) )
Explanation opens after your attempt
Correct Answer
A. ( (3n+1)(3n) )
Step 1
Concept
((3n+1)!=(3n+1)(3n)(3n-1)!), so two factors remain. Handle the indexed term as one unit.
Step 2
Why this answer is correct
The correct answer is A. ( (3n+1)(3n) ). ((3n+1)!=(3n+1)(3n)(3n-1)!), so two factors remain. Handle the indexed term as one unit.
Step 3
Exam Tip
((3n+1)!=(3n+1)(3n)(3n-1)!), इसलिए दो गुणक बचते हैं। सूचक वाले पद को एक इकाई की तरह संभालें।
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यदि ( \frac{(3n+1)!}{(3n-1)!}=156 ), तो (n) का मान क्या है?
If ( \frac{(3n+1)!}{(3n-1)!}=156 ), 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
It is ((3n+1)(3n)=156), and \(13\cdot12=156\), so (3n=12) and (n=4). Identify the form and substitute.
Step 2
Why this answer is correct
The correct answer is B. (4). It is ((3n+1)(3n)=156), and \(13\cdot12=156\), so (3n=12) and (n=4). Identify the form and substitute.
Step 3
Exam Tip
यह ((3n+1)(3n)=156) है और \(13\cdot12=156\), इसलिए (3n=12) और (n=4)। रूप पहचानकर मान रखें।
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( \frac{(n!)2 }{(n-2)!(n+2)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n!)2 }{(n-2)!(n+2)!} )?
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A \( \frac{n}{n+2} \)
B ( \frac{n(n-1)}{(n+1)(n+2)} )
C \( \frac{n-1}{n+1} \)
D \( \frac{n+1}{n-1} \)
Explanation opens after your attempt
Correct Answer
B. ( \frac{n(n-1)}{(n+1)(n+2)} )
Step 1
Concept
( \frac{n!}{(n-2)!}=n(n-1) ) and ( \frac{n!}{(n+2)!}=\frac{1}{(n+2)(n+1)} ), so the answer follows. Break the ratio into two parts.
Step 2
Why this answer is correct
The correct answer is B. ( \frac{n(n-1)}{(n+1)(n+2)} ). ( \frac{n!}{(n-2)!}=n(n-1) ) and ( \frac{n!}{(n+2)!}=\frac{1}{(n+2)(n+1)} ), so the answer follows. Break the ratio into two parts.
Step 3
Exam Tip
( \frac{n!}{(n-2)!}=n(n-1) ) और ( \frac{n!}{(n+2)!}=\frac{1}{(n+2)(n+1)} ), इसलिए उत्तर मिलता है। अनुपात को दो भागों में तोड़ें।
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यदि ( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!}=270 ), तो (n) का मान क्या है?
If ( \frac{(n+3)!}{n!}-\frac{(n+2)!}{(n-1)!}=270 ), what is the value of (n)?
#permutations
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#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 (3(n+2)(n+1)), so ((n+2)(n+1)=90). Since \(10\cdot9=90\), (n=8).
Step 2
Why this answer is correct
The correct answer is B. (8). The simplified form is (3(n+2)(n+1)), so ((n+2)(n+1)=90). Since \(10\cdot9=90\), (n=8).
Step 3
Exam Tip
सरल रूप (3(n+2)(n+1)) है, इसलिए ((n+2)(n+1)=90)। \(10\cdot9=90\), अतः (n=8)।
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(20!) के अंत में कितने शून्य होंगे?
How many zeros will be at the end of (20!)?
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#hard
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The number of ending zeros is \( \left\lfloor\frac{20}{5}\right\rfloor=4 \). To count zeros, count multiples of (5).
Step 2
Why this answer is correct
The correct answer is B. (4). The number of ending zeros is \( \left\lfloor\frac{20}{5}\right\rfloor=4 \). To count zeros, count multiples of (5).
Step 3
Exam Tip
अंतिम शून्यों की संख्या \( \left\lfloor\frac{20}{5}\right\rfloor=4 \) है। शून्य गिनने में (5) के गुणकों को गिनें।
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(30!) में (5) की अधिकतम घात क्या है?
What is the highest power of (5) in (30!)?
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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
The exponent is \( \left\lfloor\frac{30}{5}\right\rfloor+\left\lfloor\frac{30}{25}\right\rfloor=7 \). Do not forget higher multiples such as (25).
Step 2
Why this answer is correct
The correct answer is B. (7). The exponent is \( \left\lfloor\frac{30}{5}\right\rfloor+\left\lfloor\frac{30}{25}\right\rfloor=7 \). Do not forget higher multiples such as (25).
Step 3
Exam Tip
घात \( \left\lfloor\frac{30}{5}\right\rfloor+\left\lfloor\frac{30}{25}\right\rfloor=7 \) है। (25) जैसे उच्च गुणकों को न भूलें।
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(15!) को विभाजित करने वाली (3) की अधिकतम घात क्या है?
What is the highest power of (3) that divides (15!)?
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#hard
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
The exponent is \( \left\lfloor\frac{15}{3}\right\rfloor+\left\lfloor\frac{15}{9}\right\rfloor=6 \). For a prime exponent, add the quotients.
Step 2
Why this answer is correct
The correct answer is B. (6). The exponent is \( \left\lfloor\frac{15}{3}\right\rfloor+\left\lfloor\frac{15}{9}\right\rfloor=6 \). For a prime exponent, add the quotients.
Step 3
Exam Tip
घात \( \left\lfloor\frac{15}{3}\right\rfloor+\left\lfloor\frac{15}{9}\right\rfloor=6 \) है। अभाज्य घात के लिए भागफल जोड़ते हैं।
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सबसे छोटा धनात्मक (n) क्या है जिसके लिए (n!) संख्या (180) से विभाज्य हो?
What is the smallest positive (n) for which (n!) is divisible by (180)?
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A (5)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
\(180=2^2\cdot3^2\cdot5\), and (6!) contains all these factors. For divisibility, check prime factors.
Step 2
Why this answer is correct
The correct answer is B. (6). \(180=2^2\cdot3^2\cdot5\), and (6!) contains all these factors. For divisibility, check prime factors.
Step 3
Exam Tip
\(180=2^2\cdot3^2\cdot5\) है और (6!) में ये सभी गुणक मिल जाते हैं। विभाज्यता में अभाज्य गुणनखंड जांचें।
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\( \frac{21!}{19!\cdot2!}-\frac{20!}{18!\cdot2!} \) का मान क्या है?
What is the value of \( \frac{21!}{19!\cdot2!}-\frac{20!}{18!\cdot2!} \)?
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#hard
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A (19)
B (20)
C (21)
D (22)
Explanation opens after your attempt
Step 1
Concept
The first term is (210) and the second is (190), so the difference is (20). Subtract nearby terms carefully.
Step 2
Why this answer is correct
The correct answer is B. (20). The first term is (210) and the second is (190), so the difference is (20). Subtract nearby terms carefully.
Step 3
Exam Tip
पहला पद (210) और दूसरा (190) है, इसलिए अंतर (20) है। पास-पास वाले पदों में घटाव सावधानी से करें।
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\( \frac{12!}{8!\cdot4!}\div\frac{11!}{7!\cdot4!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{12!}{8!\cdot4!}\div\frac{11!}{7!\cdot4!} \)?
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A \( \frac{3}{2} \)
B \( \frac{4}{3} \)
C \( \frac{5}{4} \)
D \( \frac{6}{5} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{3}{2} \)
Step 1
Concept
The two terms are (495) and (330), so the ratio is \( \frac{3}{2} \). Convert large values into a simple ratio.
Step 2
Why this answer is correct
The correct answer is A. \( \frac{3}{2} \). The two terms are (495) and (330), so the ratio is \( \frac{3}{2} \). Convert large values into a simple ratio.
Step 3
Exam Tip
दोनों पद (495) और (330) हैं, इसलिए अनुपात \( \frac{3}{2} \) है। बड़े मानों को सरल अनुपात में बदलें।
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\( \frac{8!}{5!}+\frac{7!}{4!} \) का मान क्या है?
What is the value of \( \frac{8!}{5!}+\frac{7!}{4!} \)?
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#hard
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A (504)
B (546)
C (560)
D (588)
Explanation opens after your attempt
Step 1
Concept
The first term is \(8\cdot7\cdot6=336\) and the second is \(7\cdot6\cdot5=210\). The sum is (546).
Step 2
Why this answer is correct
The correct answer is B. (546). The first term is \(8\cdot7\cdot6=336\) and the second is \(7\cdot6\cdot5=210\). The sum is (546).
Step 3
Exam Tip
पहला पद \(8\cdot7\cdot6=336\) और दूसरा \(7\cdot6\cdot5=210\) है। योग (546) है।
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\( \frac{9!}{4!\cdot5!}-\frac{8!}{4!\cdot4!} \) का मान क्या है?
What is the value of \( \frac{9!}{4!\cdot5!}-\frac{8!}{4!\cdot4!} \)?
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#hard
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A (54)
B (56)
C (58)
D (60)
Explanation opens after your attempt
Step 1
Concept
The first term is (126) and the second is (70), so the difference is (56). Simplify each term separately.
Step 2
Why this answer is correct
The correct answer is B. (56). The first term is (126) and the second is (70), so the difference is (56). Simplify each term separately.
Step 3
Exam Tip
पहला पद (126) और दूसरा (70) है, इसलिए अंतर (56) है। हर पद को अलग सरल करें।
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यदि ( \frac{(n+5)!}{(n+2)!}=1320 ), तो (n) का मान क्या है?
If ( \frac{(n+5)!}{(n+2)!}=1320 ), 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 is ((n+5)(n+4)(n+3)=1320). Since \(12\cdot11\cdot10=1320\), (n=7).
Step 2
Why this answer is correct
The correct answer is C. (8). It is ((n+5)(n+4)(n+3)=1320). Since \(12\cdot11\cdot10=1320\), (n=7).
Step 3
Exam Tip
यह ((n+5)(n+4)(n+3)=1320) है और \(13\cdot12\cdot11=1716\) नहीं; \(12\cdot11\cdot10=1320\), इसलिए (n=7)।
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यदि ( \frac{(n+2)!}{(n-1)!}-\frac{(n+1)!}{(n-2)!}=330 ), तो (n) का मान क्या है?
If ( \frac{(n+2)!}{(n-1)!}-\frac{(n+1)!}{(n-2)!}=330 ), 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 (3n(n+1)), so (n(n+1)=110). Since \(10\cdot11=110\), (n=10).
Step 2
Why this answer is correct
The correct answer is C. (10). The simplified form is (3n(n+1)), so (n(n+1)=110). Since \(10\cdot11=110\), (n=10).
Step 3
Exam Tip
सरल रूप (3n(n+1)) है, इसलिए (n(n+1)=110)। \(10\cdot11=110\), इसलिए (n=10)।
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\( \frac{19!}{17!\cdot2!}\times\frac{2}{19} \) का मान क्या है?
What is the value of \( \frac{19!}{17!\cdot2!}\times\frac{2}{19} \)?
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A (17)
B (18)
C (19)
D (20)
Explanation opens after your attempt
Step 1
Concept
The first part is (171), and \(171\cdot\frac{2}{19}=18\). Cancel by (19) before multiplying.
Step 2
Why this answer is correct
The correct answer is B. (18). The first part is (171), and \(171\cdot\frac{2}{19}=18\). Cancel by (19) before multiplying.
Step 3
Exam Tip
पहला भाग (171) है और \(171\cdot\frac{2}{19}=18\)। गुणन से पहले (19) से कटौती करें।
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\( \frac{6!+5!+4!}{4!} \) का मान क्या है?
What is the value of \( \frac{6!+5!+4!}{4!} \)?
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A (35)
B (36)
C (37)
D (38)
Explanation opens after your attempt
Step 1
Concept
\(6!=30\cdot4!\), \(5!=5\cdot4!\), and \(4!=1\cdot4!\), so the total is (36).
Step 2
Why this answer is correct
The correct answer is A. (35). \(6!=30\cdot4!\), \(5!=5\cdot4!\), and \(4!=1\cdot4!\), so the total is (36).
Step 3
Exam Tip
\(6!=30\cdot4!\), \(5!=5\cdot4!\) और \(4!=1\cdot4!\), इसलिए कुल (36) नहीं बल्कि (36) है।
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\( \frac{10!}{8!} \), \( \frac{9!}{7!} \) से कितना अधिक है?
By how much is \( \frac{10!}{8!} \) greater than \( \frac{9!}{7!} \)?
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A (12)
B (14)
C (16)
D (18)
Explanation opens after your attempt
Step 1
Concept
The first value is (90) and the second is (72). The difference is (18).
Step 2
Why this answer is correct
The correct answer is D. (18). The first value is (90) and the second is (72). The difference is (18).
Step 3
Exam Tip
पहला मान (90) और दूसरा (72) है। अंतर (18) है।
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( \frac{(n+4)!}{n!} ) के विस्तार में कौन सा गुणक शामिल नहीं होगा?
Which factor will not appear in the expansion of ( \frac{(n+4)!}{n!} )?
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A (n+1)
B (n+2)
C (n+3)
D (n)
Explanation opens after your attempt
Step 1
Concept
The expansion is ((n+4)(n+3)(n+2)(n+1)), so (n) is not included. Cancellation stops at the denominator (n!).
Step 2
Why this answer is correct
The correct answer is D. (n). The expansion is ((n+4)(n+3)(n+2)(n+1)), so (n) is not included. Cancellation stops at the denominator (n!).
Step 3
Exam Tip
विस्तार ((n+4)(n+3)(n+2)(n+1)) है, इसलिए (n) शामिल नहीं है। हर वाले (n!) तक कटौती हो जाती है।
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(n=4) होने पर ( \frac{(n+4)!}{n!} ) का मान क्या है?
When (n=4), what is the value of ( \frac{(n+4)!}{n!} )?
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A (1680)
B (1520)
C (1480)
D (1340)
Explanation opens after your attempt
Step 1
Concept
Putting (n=4) gives \( \frac{8!}{4!}=8\cdot7\cdot6\cdot5=1680 \). Understand the general form before substituting.
Step 2
Why this answer is correct
The correct answer is A. (1680). Putting (n=4) gives \( \frac{8!}{4!}=8\cdot7\cdot6\cdot5=1680 \). Understand the general form before substituting.
Step 3
Exam Tip
(n=4) रखने पर \( \frac{8!}{4!}=8\cdot7\cdot6\cdot5=1680 \) मिलता है। मान रखने से पहले सामान्य रूप समझें।
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यदि \( \frac{(n+1)!}{(n-1)!}-\frac{n!}{(n-2)!}=24 \), तो \(n\) का मान क्या है?
If \( \frac{(n+1)!}{(n-1)!}-\frac{n!}{(n-2)!}=24 \), what is the value of \(n\)?
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A (10)
B (11)
C (12)
D (13)
Explanation opens after your attempt
Step 1
Concept
The simplified form is (2n), so \(2n=24\). Hence \(n=12\).
Step 2
Why this answer is correct
The correct answer is C. (12). The simplified form is (2n), so \(2n=24\). Hence \(n=12\).
Step 3
Exam Tip
सरल रूप (2n) है, इसलिए \(2n=24\)। अतः \(n=12\)।
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\( \frac{22!}{20!\cdot2!}-\frac{21!}{20!} \) का मान क्या है?
What is the value of \( \frac{22!}{20!\cdot2!}-\frac{21!}{20!} \)?
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A (210)
B (220)
C (231)
D (240)
Explanation opens after your attempt
Step 1
Concept
The first term is (231) and the second is (21). The difference is (210).
Step 2
Why this answer is correct
The correct answer is A. (210). The first term is (231) and the second is (21). The difference is (210).
Step 3
Exam Tip
पहला पद (231) और दूसरा (21) है। अंतर (210) है।
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\(\frac{17!}{14!}\div\frac{8!}{5!} \) का सरल मान क्या है?
What is the simplified value of \( \frac{17!}{14!}\div\frac{8!}{5!} \)?
#permutations
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A \( \frac{17}{2} \)
B \( \frac{85}{8} \)
C \( \frac{255}{28} \)
D \( \frac{17}{4} \)
Explanation opens after your attempt
Correct Answer
C. \( \frac{255}{28} \)
Step 1
Concept
The value is \( \frac{17\cdot16\cdot15}{8\cdot7\cdot6}=\frac{255}{28} \). In division, write three factors on both sides.
Step 2
Why this answer is correct
The correct answer is C. \( \frac{255}{28} \). The value is \( \frac{17\cdot16\cdot15}{8\cdot7\cdot6}=\frac{255}{28} \). In division, write three factors on both sides.
Step 3
Exam Tip
मान \( \frac{17\cdot16\cdot15}{8\cdot7\cdot6}=\frac{255}{28} \) है। भाग में दोनों तरफ तीन-तीन गुणक लिखें।
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यदि \(\frac{(n+2)!}{(n-2)!}=3024\), तो (n) का मान क्या है?
If \(\frac{(n+2)!}{(n-2)!}=3024 \), 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
It is \((n+2)(n+1)n(n-1)=3024\). Since \(9\cdot8\cdot7\cdot6=3024\), (n=7).
Step 2
Why this answer is correct
The correct answer is B. (7). It is \((n+2)(n+1)n(n-1)=3024\). Since \(9\cdot8\cdot7\cdot6=3024\), (n=7).
Step 3
Exam Tip
यह \((n+2)(n+1)n(n-1)=3024\) है। \(9\cdot8\cdot7\cdot6=3024\), इसलिए (n=7)।
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\(\frac{10!}{2!\cdot8!}\times\frac{8}{9}\) का मान क्या है?
What is the value of \( \frac{10!}{2!\cdot8!}\times\frac{8}{9} \)?
#permutations
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#hard
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A (20)
B (30)
C (40)
D (50)
Explanation opens after your attempt
Step 1
Concept
The first part is (45), and \(45\cdot\frac{8}{9}=40\). Cancel by (9) first.
Step 2
Why this answer is correct
The correct answer is C. (40). The first part is (45), and \(45\cdot\frac{8}{9}=40\). Cancel by (9) first.
Step 3
Exam Tip
पहला भाग (45) है और \(45\cdot\frac{8}{9}=40\)। पहले (9) से कटौती करें।
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\(\frac{(n+6)!}{(n+3)!}\) का सरल रूप क्या है?
What is the simplified form of \(\frac{(n+6)!}{(n+3)!}\)?
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A ( (n+6)(n+5) )
B ( (n+6)(n+5)(n+4) )
C ( (n+5)(n+4)(n+3) )
D ( (n+6)(n+4)(n+3) )
Explanation opens after your attempt
Correct Answer
B. ( (n+6)(n+5)(n+4) )
Step 1
Concept
\((n+6)!=(n+6)(n+5)(n+4)(n+3)!\), so three factors remain. Do not forget to expand up to the denominator.
Step 2
Why this answer is correct
The correct answer is B. ( (n+6)(n+5)(n+4) ). \((n+6)!=(n+6)(n+5)(n+4)(n+3)!\), so three factors remain. Do not forget to expand up to the denominator.
Step 3
Exam Tip
\((n+6)!=(n+6)(n+5)(n+4)(n+3)!\), इसलिए तीन गुणक बचते हैं। हर तक विस्तार करना न भूलें।
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\( \frac{14!}{10!\cdot4!}-\frac{13!}{10!\cdot3!} \) का मान क्या है?
What is the value of \( \frac{14!}{10!\cdot4!}-\frac{13!}{10!\cdot3!} \)?
#permutations
#combinations
#factorial
#hard
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A (689)
B (715)
C (742)
D (786)
Explanation opens after your attempt
Step 1
Concept
The first term is (1001) and the second is (286), so the difference is (715). In such questions, calculate both terms separately and subtract.
Step 2
Why this answer is correct
The correct answer is B. (715). The first term is (1001) and the second is (286), so the difference is (715). In such questions, calculate both terms separately and subtract.
Step 3
Exam Tip
पहला पद (1001) और दूसरा (286) है, इसलिए अंतर (715) है। ऐसे प्रश्नों में दोनों पद अलग-अलग निकालकर घटाएं।
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( \frac{\frac{(n+4)!}{n!}}{\frac{(n+3)!}{(n-1)!}} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{\frac{(n+4)!}{n!}}{\frac{(n+3)!}{(n-1)!}} )?
#permutations
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#hard
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A \( \frac{n}{n+4} \)
B \( \frac{n+3}{n} \)
C \( \frac{n+4}{n+1} \)
D \( \frac{n+4}{n} \)
Explanation opens after your attempt
Correct Answer
D. \( \frac{n+4}{n} \)
Step 1
Concept
After canceling the common factors in numerator and denominator, \( \frac{n+4}{n} \) remains. First convert large ratios into products.
Step 2
Why this answer is correct
The correct answer is D. \( \frac{n+4}{n} \). After canceling the common factors in numerator and denominator, \( \frac{n+4}{n} \) remains. First convert large ratios into products.
Step 3
Exam Tip
ऊपर और नीचे के समान गुणक कटने पर \( \frac{n+4}{n} \) बचता है। बड़े अनुपात को पहले गुणनफल में बदलें।
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(14!) को विभाजित करने वाली (2) की अधिकतम घात क्या है?
What is the highest power of (2) that divides (14!)?
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#factorial
#hard
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A (9)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
The exponent is \( \left\lfloor\frac{14}{2}\right\rfloor+\left\lfloor\frac{14}{4}\right\rfloor+\left\lfloor\frac{14}{8}\right\rfloor=11 \). Add all quotients while finding a prime exponent.
Step 2
Why this answer is correct
The correct answer is C. (11). The exponent is \( \left\lfloor\frac{14}{2}\right\rfloor+\left\lfloor\frac{14}{4}\right\rfloor+\left\lfloor\frac{14}{8}\right\rfloor=11 \). Add all quotients while finding a prime exponent.
Step 3
Exam Tip
घात \( \left\lfloor\frac{14}{2}\right\rfloor+\left\lfloor\frac{14}{4}\right\rfloor+\left\lfloor\frac{14}{8}\right\rfloor=11 \) है। अभाज्य घात निकालते समय सभी भागफल जोड़ें।
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सबसे छोटा धनात्मक (n) क्या है जिसके लिए (n!) संख्या (1008) से विभाज्य हो?
What is the smallest positive (n) for which (n!) is divisible by (1008)?
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
\(1008=2^4\cdot3^2\cdot7\), and (7!) contains all these factors. For divisibility, satisfy all prime factor requirements.
Step 2
Why this answer is correct
The correct answer is B. (7). \(1008=2^4\cdot3^2\cdot7\), and (7!) contains all these factors. For divisibility, satisfy all prime factor requirements.
Step 3
Exam Tip
\(1008=2^4\cdot3^2\cdot7\) है और (7!) में ये सभी गुणक मिल जाते हैं। विभाज्यता में अभाज्य गुणनखंडों की जरूरत पूरी करें।
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( \frac{(n+4)!+(n+3)!}{(n+2)!} ) का सरल रूप क्या है?
What is the simplified form of ( \frac{(n+4)!+(n+3)!}{(n+2)!} )?
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A ( (n+3)(n+5) )
B ( (n+4)(n+3) )
C ( (n+3)2 )
D ( (n+2)(n+5) )
Explanation opens after your attempt
Correct Answer
A. ( (n+3)(n+5) )
Step 1
Concept
Taking ((n+3)!) common gives ((n+3)!(n+5)) in the numerator. Dividing gives ((n+3)(n+5)).
Step 2
Why this answer is correct
The correct answer is A. ( (n+3)(n+5) ). Taking ((n+3)!) common gives ((n+3)!(n+5)) in the numerator. Dividing gives ((n+3)(n+5)).
Step 3
Exam Tip
((n+3)!) सामान्य लेने पर ऊपर ((n+3)!(n+5)) बनता है। भाग देने पर ((n+3)(n+5)) मिलता है।
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यदि ( \frac{(n+4)!+(n+3)!}{(n+2)!}=120 ), तो (n) का मान क्या है?
If ( \frac{(n+4)!+(n+3)!}{(n+2)!}=120 ), 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
The simplified form is ((n+3)(n+5)), so ((n+3)(n+5)=120). Since \(10\cdot12=120\), (n=7).
Step 2
Why this answer is correct
The correct answer is B. (7). The simplified form is ((n+3)(n+5)), so ((n+3)(n+5)=120). Since \(10\cdot12=120\), (n=7).
Step 3
Exam Tip
सरल रूप ((n+3)(n+5)) है, इसलिए ((n+3)(n+5)=120)। \(10\cdot12=120\), इसलिए (n=7)।
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\( \frac{12!}{9!}-2\cdot\frac{11!}{8!} \) का मान क्या है?
What is the value of \( \frac{12!}{9!}-2\cdot\frac{11!}{8!} \)?
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A (610)
B (630)
C (650)
D (660)
Explanation opens after your attempt
Step 1
Concept
The first term is (1320) and the second is \(2\cdot330=660\), so the difference is (660). Complete the multiplication in the coefficient term first.
Step 2
Why this answer is correct
The correct answer is D. (660). The first term is (1320) and the second is \(2\cdot330=660\), so the difference is (660). Complete the multiplication in the coefficient term first.
Step 3
Exam Tip
पहला पद (1320) और दूसरा \(2\cdot330=660\) है, इसलिए अंतर (660) है। गुणांक वाले पद में पहले गुणा पूरा करें।
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