Problem 5 · The coefficient of x⁵
Only two contributions: the binomial and the $-8x^{5}$ term.
Integer answer, at most 4 digitsWhat is the coefficient of $x^{5}$ in the expansion of $(1 - x^{5})^{8} \cdot (1 + 2x)^{6}$?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: from $(1-x^{5})^{8}$ only the terms $1$ and $-8x^{5}$ can contribute to $x^{5}$ (the next ones already have degree $10$ or more).
Term $1 \times$ (coefficient of $x^{5}$ in $(1+2x)^{6}$):
$$\binom{6}{5} 2^{5} = 6 \cdot 32 = 192.$$
Term $-8x^{5} \times$ (constant term of $(1+2x)^{6}$, which is $1$): $-8$.
$$192 - 8 = 184.$$
Answer: 184