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Mathematics Question on binomial expansion formula

The number (492−4)(492−49)(49^2 -4) (49^2 -49) is divisible by ..................

A

6!

B

5!

C

7!

D

9!

Answer

5!

Explanation

Solution

Given, (492−4)(493−49)\left(49^{2}-4\right)\left(49^{3}-49\right)
=[(49)2−(2)2][(49)2−1]⋅49=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49
=(49+2)(49−2)(49+1)(49−1)⋅49=(49+2)(49-2)(49+1)(49-1) \cdot 49
=51⋅47⋅50⋅48⋅49=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49
=(51⋅50⋅49⋅48⋅47)=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)
Which is the product of five consecutive integers and hence it divisible by 5!5 !.