Question

Consider the expansion of (5p + 2q)^6. Determine the coefficients for the terms with the powers of p and q shown.

Answer 1

Answer:

Remember, the expansion of $(x+y)^n$ is $(x+y)^n=\sum_{k=0}^n \binom{n}{k}x^{n-k}y^k$, where $\binom{n}{k}=\frac{n!}{(n-k)!k!}$.

Then,

$(5p+2q)^6=\sum_{k=0}^6\binom{6}{k}(5p)^{6-k}(2q)^k=\sum_{k=0}^6\binom{6}{k}5^{6-k}2^k p^{6-k}q^k$

Then, the coefficient of the term $p^{6-k}q^k$ is $\binom{6}{k}5^{6-k}2^k$

a) since 6-k=2, then k=4. So the coefficient of $p^2q^4$ is

$\binom{6}{4}5^{6-4}2^4=15*5^2*2^4=15*25*16=6000$

b) since 6-k=5, then k=1. So, the coefficient of $p^5q$ is

$\binom{6}{1}5^{6-1}2^1=6*5^5*2=37500$

c) since 6-k=3, then k=3. So, the coefficient of $p^3q^3$ is

$\binom{6}{3}5^{6-3}2^3=20*5^3*8=20000$