### Factorials

**factorial**is the product of all natural numbers less than or equal to the indicated natural number. The symbol for a factorial is an exclamation point.

### Permutations

A **permutation** is the selection and arrangement of objects in a certain order. For example, there are many ways five siblings can arrange themselves in a row of five seats at the movie theater. The order in which the siblings are seated is different in each arrangement, so the situation involves permutations. To determine the number of unique ways the siblings can be ordered, use a factorial.

When there are more objects to choose from than objects that will be arranged, a formula is used to calculate the number of permutations.

The number of permutations of $n$ objects chosen $r$ at a time is:Determine whether the situation involves permutations. If so, find the values for $n$ and $r$.

The order of the students chosen for the positions matters because only one student can be chosen for each position. For instance, the student chosen for president is different from a student chosen for vice president. The situation involves permutations.

There are $n=12$ students to choose from. There are $r=4$ students who will be chosen for the available positions.

### Combinations

A **combination** is the selection of objects in groups where order does not matter. There are many ways that three different types of fruit can be selected from five types of fruit available at a farm stand. When order does not matter, selecting an apple, a peach, and strawberries versus selecting strawberries, a peach, and an apple results in the same three different types of fruits. The order in which the different types of fruit are selected does not matter, so the situation involves combinations.

Determine whether the situation involves permutations or combinations. Then find the values for $n$ and $r$.

Each student will have the same role on the council, so the order in which the students are selected does not matter. The situation involves combinations.

There are $n=80$ students vying to be on the council. There are $r=6$ students who will be selected to be on the council.

Determine whether the situation involves permutations or combinations. Then find the values for $n$ and $r$.

Rearranging the letters means putting them in a different order, so the order of the letters matters. The situation involves permutations.

There are $n=9$ letters in the word. There are $r=5$ letters that will be rearranged for each permutation.

Some of the letters are being selected and put into a different order.

Substitute values in the permutation formula and simplify.