Permutations

When the elements of a set are arranged in a definite order, the arrangement is called a permutation of the elements.  The number of permutations of $n$ objects is $n!$

The number of possible orderings of $m$ objects taken from a set of $n$ is given by:

$\begin{array}{l}{}_{n}P{}_m=n\times \left(n-1\right)\times \left(n-2\right)\times \cdots \times \left(n-m+1\right)\\ =\frac{n!}{\left(n-m\right)!}\end{array}$

That is, count backwards starting from $n$ , writing down the numbers as you count, until you've written down $m$ numbers. Then multiply them all together.

Using the permutations formula, we have:

${}_{5}P{}_3=5\times 4\times 3=60$

To see why this works, name the shows A, B, C, D, and E, and make a list:

In this case, there are $5$ choices for the first program, $4$ choices for the second program, and $3$ choices for the last program. So the answer is:

${}_{5}P{}_3=5\times 4\times 3=60$

If there were $8$ programs and $4$ time slots, we would have:

${}_{8}P{}_4=8\times 7\times 6\times 5=1680$