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

Answer:

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 $\binom{8}{4}=70$