State the factorial formula for \(\binom{n}{r}\).

Evaluate \(\binom{6}{0}\).

Evaluate \(\binom{5}{2}\).

Use symmetry to rewrite \(\binom{12}{9}\) with a smaller lower number, then evaluate it if possible.

Evaluate \(\binom{7}{3}\).

How many ways are there to choose \(2\) students from \(8\) students when order does not matter?

Find the row \(5\) entries of Pascal's triangle using binomial coefficients.

Use Pascal's recurrence to compute \(\binom{6}{3}\) from row \(5\).

Show that \(\binom{n}{1}=n\).

A group has \(6\) labelled objects. Compare the number of ordered selections of \(3\) objects with unordered selections of \(3\) objects.

Explain combinatorially why \(\binom{n}{r}=\binom{n}{n-r}\).

Explain why the edge entries of Pascal's triangle are \(1\).

Prove Pascal's recurrence \(\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}\) by separating cases for one distinguished object.

Evaluate \(\binom{10}{4}\) efficiently using cancellation.

Find all \(r\) such that \(\binom{8}{r}=\binom{8}{3}\).

A committee of \(4\) is chosen from \(5\) physicists and \(3\) mathematicians. How many committees have exactly \(2\) physicists?

How many ways can \(5\) objects be chosen from \(9\) if at least one of two specified objects must be included?

Diagnose the error: \(\binom{7}{2}=\frac{7!}{2!}=2520\).

Prove algebraically that \(\binom{n}{r}=\binom{n}{n-r}\).

Find the number of subsets of a set with \(n\) elements by using binomial coefficients, and explain the result.