Question

The coefficient of x⁴⁹ in the product

(x - 1)(x - 3) ... (x - 99) is

(a) - 99² (b) 1 (c) - 2500 (d) - 50

Answer 1

(x−a)(x−b) =x²−(a+b)x+ab

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(x−a)(x−b)(x−c) =
x³-(a+b+c)x²+(ab+bc+ca)x−abc

So, can you guess the pattern for product of 4 binomials?

(x−a)(x−b)(x−c)(x−d) =x⁴−(a+b+c+d)x³+(ab+bc+cd+da)x²−(abc+bcd+cda+abd)x+abcd

So, for the given product

(x−1)(x−3)(x−5)⋯⋯(x−99) =x^50−(1+3+5+⋯+99)x^49+(1∗3+1∗5+1∗7⋯+1∗99+3∗5+3∗7+⋯3∗99+5∗7+⋯+97∗99)x^48+⋯

Clearly, the answer is = - [Sum of odd numbers from 1 to 50] =−502(1+99) =−2500.