Question

Determine if the situation describes a permutation or a combination. Justify your reasoning. Then solve the problem.

Denise selects three different toppings for a pizza. If there are 9 different toppings from which to choose, how many different pizzas can be made?

Answer 1

The situation describes a combination rather than a permutation. The key distinction is that the order of the toppings does not matter. When selecting toppings for a pizza, it is the selection of items that counts, not the arrangement of those items. Therefore, we are interested in finding how many different groups of three toppings can be formed from a set of nine.

To solve the problem of how many different combinations of three toppings can be selected from nine different toppings, we can use the combination formula:

\[
C(n, k) = \frac{n!}{k!(n-k)!}
\]

where \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and \( ! \) denotes factorial.

In this case:
- \( n = 9 \) (the total number of toppings)
- \( k = 3 \) (the number of toppings being chosen)

Plugging the values into the formula, we get:

\[
C(9, 3) = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!}
\]

Calculating the factorials, we find:

\[
9! = 9 \times 8 \times 7 \times 6!
\]
Thus,

\[
C(9, 3) = \frac{9 \times 8 \times 7 \times 6!}{3! \times 6!}
\]

The \( 6! \) cancels out:

\[
C(9, 3) = \frac{9 \times 8 \times 7}{3!}
\]

Calculating \( 3! \):

\[
3! = 3 \times 2 \times 1 = 6
\]

Now, substituting back into our equation:

\[
C(9, 3) = \frac{9 \times 8 \times 7}{6}
\]

Calculating the numerator:

\[
9 \times 8 = 72
\]
\[
72 \times 7 = 504
\]

Now, divide by 6:

\[
C(9, 3) = \frac{504}{6} = 84
\]

Thus, the number of different pizzas that can be made with three different toppings from nine options is **84**.