Question

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

a. p^2q^4
b. p^5q
c. p^3q^3

Answer 1

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}=(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`