Question

How many ways can 3 students sit in a row of 20 chairs for a photograph?

Answer 1

Since we want to create groups of 3 students among 20 possible spots, and the order of the elements (students) inside each group of 3 is important (the order of students change the group and the photograph), we have a permutation problem.

The formula to calculate a permutation of n choose p is:

$P(n,p)=\frac{n!}{(n-p)!}$

For this problem, let's use n = 20 and p = 3, so we have:

$\begin{gathered} P(20,3)=\frac{20!}{(20-3)!} \\ P(20,3)=\frac{20!}{17!} \\ P(20,3)=\frac{20\cdot19\cdot18\cdot17!}{17!} \\ P(20,3)=20\cdot19\cdot18 \\ P(20,3)=6840 \end{gathered}$

So there are 6840 different ways the 3 students can sit in the chairs.