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Handout 2.2 Ubiquitous Nature of Binomial Coefficients
Activity 2.2.1 .
Find a simple formula for
\(\ds\sum_{k=1}^{n} k\binom{n}{k}^{2}\text{.}\) By a simple formula, I mean one without a summation in it. One way to do this is to think about something that’s being counted and find a "better" way to count it.
Hint .
Think of a committee (or team) with a chair (or captain).
Remember that
\(\displaystyle \binom{n}{k}= \binom{n}{n-k}\)
Peer instruction questions 1–4.
Activity 2.2.2 .
We want to count the number of solutions to
\begin{equation*}
x_1+x_2+x_3+x_4+x_5 = 37
\end{equation*}
where each \(x_i\) is an integer. For each of the following scenarios, determine the number of solutions.
(a)
(b)
(c) Each
\(x_i \geq 0\) and
\(x_3\gt 4\)
(d) Each
\(x_i \gt 0\) but change
\(= 37\) to
\(\leq 37\text{.}\)
(e) Each
\(x_i \gt 0\) but change
\(= 37\) to
\(\lt 37\text{.}\)
