Question

Coefficient of `x^(11)` in the expansion of `(1+x^2)(1+x^3)^7(1+x^4)^(12)` is 1051 b. 1106 c. 1113 d. 1120
A. 1051
B. 1106
C. 1113
D. 1120

Answer 1

Coefficient of `x^r` in `(1+x)^n` is `.^nC_r`.
In this type of questions, we find different composition of terms where product will give us `x^11`.
Now , consider the following cases for `x^11` in
`(1+x^2)^4(1+x^3)^7(1+x^4)^12`.
Coefficient of `x^0x^3x^8`, Coefficient of `x^2x^9x^0`
Coefficient of `x^4x^3x^4`, Coefficient of `x^8 x^3 x^0`
`=.^4C_0 xx .^7C_1 xx .^12C_2 xx .^4C_1 xx .^7C_3 xx .^12C_0+.^C_0+.^4C_2 xx .^7C_1xx .^12C_1 +.^4C_4 xx .^7C_1 xx .^12C_0`
`=462 +140 +504+7=1113`.