Explanation
{\textbf{Step -1: Identify binomial coefficients and number of terms in a binomial expansion.}}
(x^2-x-2)^5=(x^2-2x+x-2)^5
=(x(x-2)+1(x-2))^5
=((x-2)(x+1))^5
=(x-2)^5(x+1)^5
=[^{5}C_{0}x^5+^{5}C_{1}x^4.(-2)+^{5}C_{2}x^3.(-2)^2+^{5}C_{3}x^2.(-2)^3+^{5}C_{4}x.(-2)^4+(-2)^5]
\times[^{5}C_{0}x^5+^{5}C_{1}x^4+^{5}C_{2}x^3+^{5}C_{3}x^2+^{5}C_{4}x+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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