Question

Use the binomial formula to find the coefficient of the q^4 p^17

Answer 1

Recal the binomial theorem:

$\displaystyle (a+b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k$

Then

$\displaystyle (2q+p)^{21} = \sum_{k=0}^{21} \binom{21}k (2q)^{21-k} p^k = \sum_{k=0}^{21} \binom{21}k 2^{21-k} q^{21-k} p^k$

We get the q⁴p¹⁷ term when k = 17, and its coefficient would be

$\dbinom{21}{17} 2^{21-17} = \dfrac{21!}{17!(21-17)!} 2^4 = \dfrac{21\cdot20\cdot19\cdot18}{4\cdot3\cdot2\cdot1}\cdot2^4 = \boxed{95,760}$