Question Bank - Mathematics

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Consider the following statements:1. \(\frac{n!}{3!}\) is divisible by 6, where n > 32. \(\frac{n!}{3!}+3 \) is divisible by 7, where n > 3Which of the above statements is/are correct?

A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2

Solution:

Concept:Factorial of a non-negative integer is the multiplication of all integers smaller than or equal to n.For example factorial of 6 (6!) is 6 × 5 × 4 × 3 × 2 × 1Calculation:Statement I: \(\frac{n!}{3!}\) is divisible by 6, where n > 3Let n = 4? \(\displaystyle \frac{n!}{3!}=\frac{4!}{3!}=4\)4 is not divisible by 6.Statement I is incorrect.Statement I: \(\frac{n!}{3!}+3 \) is divisible by 7, where n > 3Let n = 5? \(\displaystyle \frac{n!}{3!}+3=\frac{5!}{3!}+3=\frac{5×4×3!}{3!}+3=20+3=23\)23 is not divisible by 7.Statement II is incorrect.? Neither statement I nor II is correct.

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