Explanation
Step -1: Identify binomial coefficients and number of terms in a binomial expansion.
(x2−x−2)5=(x2−2x+x−2)5
=(x(x−2)+1(x−2))5
=((x−2)(x+1))5
=(x−2)5(x+1)5
=[5C0x5+5C1x4.(−2)+5C2x3.(−2)2+5C3x2.(−2)3+5C4x.(−2)4+(−2)5]
×[5C0x5+5C1x4+5C2x3+5C3x2+5C4x+1]
\therefore\text{coefficient of }x^5\text{ in the expansion of the product }(x-2)^5(x+1)^5
=-2^5+1+^5C_2 .^5C_3(-2)^3+^5C_3 .^5C_2(-2)^2+^5C_4 .^5C_1(-2)^1+^5C_1 .^5C_4(-2)^4
=-32+1-800+400-50+400
=-81
{\textbf{Hence, option C is correct.}}
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