Question

Given 8 different toppings to choose from, how many 3-topping pizzas are possible?

336

78

24

56

Answer 1

To determine how many 3-topping pizzas can be made from 8 different toppings, we need to calculate the number of combinations of 3 toppings that can be chosen from 8. This is done using the combination formula:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

In this case, \( n = 8 \) (the total number of toppings) and \( r = 3 \) (the number of toppings to choose).

Calculating \( \binom{8}{3} \):

\[
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \times 5!}
\]

Since \( 8! = 8 \times 7 \times 6 \times 5! \), we can simplify the calculation:

\[
\binom{8}{3} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3!}
\]

Now we calculate \( 3! \):

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

Now we can substitute \( 3! \) back into the formula:

\[
\binom{8}{3} = \frac{8 \times 7 \times 6}{6}
\]

This simplifies to:

\[
8 \times 7 = 56
\]

Therefore, the total number of different 3-topping pizzas possible is:

\[
\boxed{56}
\]