How many permutations of 5 items taken 2 at a time?

Problem: Find the number of permutations of 5 items taken 2 at a time, denoted as $P(5, 2)$ or $^{5}P_{2}$.

Steps

1. Apply the permutation formula
   $$P(n, r) = \frac{n!}{(n-r)!}$$
2. Substitute $n = 5$ and $r = 2$
   $$P(5, 2) = \frac{5!}{(5-2)!}$$
3. Simplify the denominator
   $$P(5, 2) = \frac{5!}{3!}$$
4. Expand the factorials
   $$P(5, 2) = \frac{5 \times 4 \times 3!}{3!}$$
5. Cancel the common term $3!$
   $$P(5, 2) = 5 \times 4$$
6. Multiply
   $$P(5, 2) = 20$$