The coefficient of `x^70` in the product `(x-1)(x^2-2)(x^3-3)....(x^12-12)` is

Options

A. `4`
B. `6`
C. `8`
D. `12`

Answer

A

Explanation

`(a)` The highest power of `x=1+2+3+….+12=78` to get coefficient of `x^(70)`, we have to omit the factors containing `x^(8)`
`(1)` Product of the constant terms of `(x-1)(x^(7)-7)=7`
`(2)` Product of the constant terms of `(x^(2)-2)(x^(6)-6)=12`
`(3)` Product of the constant terms of `(x^(3)-3)(x^(5)-5)=15`
`(4)` Product of the constant terms of `(x-1)(x^(2)-2)(x^(5)-5)=-10`
`(5)` Product of the constant terms of `(x-1)(x^(3)-3)(x^(4)-4)=-12`
Required coefficient `=7+12+15-10-12-8`
`=34-30=4`