Online Course Discussion Forum

MC II-B HW Questions

Hello,

For 6.26, I can see the pattern and I know that I should use the binomial theorem, but I don't really know how I should apply the binomial theorem here.

For 6.27-28, I don't how to start them.

For 6.30, I'm assuming you use the binomial theorem and then use symmetry, but I'm not sure how to apply them.

Thanks! And Happy (early) MLK Day.

On 26, you know that each term of the expansion of $(a+b)^n$ looks like $\displaystyle \binom{n}{k} a^kb^{n-k}$. The trick is trying to figure out what $a$ and $b$ are. Note each of the binomial coefficients is being multiplied by a power of $2$. This should give you a clue to find $a$ and $b$.

For 27, this may help: $$k\binom{n}{k} = \dfrac{k\cdot n!}{k! (n-k)!} = \dfrac{n!}{(k-1)!(n-k)!}= n\dfrac{(n-1)!}{(k-1)!((n-1)-(k-1))!} = \binom{n-1}{k-1},$$ then shift the index of the sum. Similar to example 6.7.

On 28, you want to use the multinomial theorem, which is pretty similar to the binomial theorem but with more than two terms. You can find it at the beginning of the chapter in the class materials, but I'll leave it here too: Let $n$ be a positive integer, then the coefficient of $a_1^{k_1}a_2^{k_2}\cdots a_j^{k_j}$ (where $k_1+k_2+\cdots +k_j = n$) in $(a_1 + a_2 + \cdots + a_j)^n$ is $\displaystyle \frac{n!}{(k_1 !)\cdot(k_2 !)\cdots (k_j !)}$.

For 30, try looking at the binomial expansion of $(1-1)^n$. What would you change to that so it matches the terms you have in the problem?