Question

Find the coefficients of the terms below in the expansion of (uu+vv)8. Explain your reasoning.

a. u2v6
b. u3v5
c. u4v4

Answer 1

Remember, the expansion of
`(u+v)^n=\sum_(k=0)^n\binom{n}{k}u^(n-k)v^k`.

Then,

`(u+v)^8=\sum_(k=0)^8\binom{8}{k}u^(8-k)v^k`

a) since
`u^(8-k)=u^2`, then
`8-k=2, k=6`. Therefore, the coefficient of
`u^2v^6` is
`\binom{8}{6}=28`

b)
`u^3v^5=u^(8-k)v^k`. Then k=5 and the coefficient of the term is
`\binom{8}{5}=56`

c)
`u^4v^4`, then k=4 and the coefficient of the term is
`\binom{8}{4}=70`