Question

The coefficient of ${x^8}$ in the polynomial $\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)....\left( {x - 10} \right)$ is:
(A) $1025$
(B) $1240$
(C) $1320$
(D) $1440$

Answer 1

Hint : The given question requires us to find the coefficient of ${x^8}$ in the polynomial given to us. The term consisting ${x^8}$ in the polynomial $\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)....\left( {x - 10} \right)$ will have a coefficient that is the sum of the product of two constant terms taken at a time because in order to get a term consisting ${x^8}$ , we have to multiply the x terms from $8$ brackets and constant terms from $2$ brackets.

Complete step by step solution:
So, we have to find the coefficient of ${x^8}$ in the polynomial $\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)....\left( {x - 10} \right)$ . So, we observe that in order to find the coefficient of a term consisting of ${x^8}$ , we have to first observe how any such term generates and is its coefficient determined.
So, we observe that any term consisting of ${x^8}$ is formed when the x terms are multiplied from $8$ brackets and constant terms from $2$ brackets. So, on observing the rule and pattern, we can determine the coefficient of ${x^8}$ in the polynomial given to us.
Coefficient of ${x^8}$ $= \left( {1 \times 2 + 1 \times 3 + ...1 \times 10} \right) + \left( {2 \times 3 + 2 \times 4 + ...2 \times 10} \right) + ... + \left( {8 \times 9 + 8 \times 10} \right) + \left( {9 \times 10} \right)$
On simplifying the expression, we get,
$\Rightarrow 1\left( {2 + 3 + 4 + ...10} \right) + 2\left( {3 + 4 + 5 + ...10} \right) + .... + 8\left( {9 + 10} \right) + 9\left( {10} \right)$
So computing the sum given in the brackets, we get,
$\Rightarrow 1\left( {54} \right) + 2\left( {52} \right) + 3\left( {49} \right) + 4\left( {45} \right) + 5\left( {40} \right) + 6\left( {34} \right) + 7\left( {27} \right) + 8\left( {19} \right) + 9\left( {10} \right)$
$\Rightarrow 54 + 104 + 147 + 180 + 200 + 204 + 189 + 152 + 90$
$\Rightarrow 1320$
So, the coefficient of ${x^8}$ in the polynomial $\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)....\left( {x - 10} \right)$ is $1320$ .
So, option (C) is correct.
So, the correct answer is “Option C”.

Note : The given question requires us to have thorough knowledge of applications of binomial theorem and employ our analytical and logical reasoning to solve the given problem. The formula for sum of terms of an AP must be remembered in order to solve the given problem.