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Hoochie [10]
2 years ago
6

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

Mathematics
1 answer:
Nata [24]2 years ago
3 0

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}

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