The coefficient of x^{99} in the expansion of (x-1)(x-2)(x-3)(x-100) is

Correct Answer is : −5050-5050−5050

Correct Answer is : −5050-5050−5050

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UPSC NDA II 2015 Mathematics Paper

Question : 113 of 120

Marks: +1, -0

x^{99}

(x-1)(x-2)(x-3)\dots(x-100)

Solution:

Concept:

The coefficient of

x^{99}

equals the negative sum of all the constant terms (the roots

1

100

Explanation:

The product

(x-1)(x-2)\ldots(x-100)

expands to

x^{100} - (1+2+\cdots+100)x^{99} + \ldots

The term

x^{99}

comes from selecting

x

from 99 factors and the constant from one factor.

Therefore, the coefficient of

x^{99}

-1 \times (1+2+\cdots+100)

Sum of first 100 natural numbers:

1+2+\cdots+100 = \frac{100 \times 101}{2} = 5050

Hence the required coefficient is

-5050

Answer:

-5050

corresponds to option C.

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