What is the coefficient of (a^{3}b^{5}) in ((a+b)^{8})?

Correct answer: 56

In the binomial expansion, the general term is \(\binom{8}{r}a^{8-r}b^r\). To obtain \(a^3b^5\), take \(r=5\); hence the coefficient is \(\binom{8}{5}=\binom{8}{3}=56\). Although one may choose the positions of either the three \(a\)'s or the five \(b\)'s, both counts are equal and refer to the same term. Exam tip: first check that the exponents add up to the binomial power; here, \(3+5=8\).