To determine how many different three-topping pizzas you can get from a pizza shop that offers 30 different toppings, you need to calculate the number of combinations of 3 toppings from the 30 available.
The number of ways to choose 3 toppings out of 30 without regard to order is given by the combination formula:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items (toppings), which is 30.
- [tex]\( k \)[/tex] is the number of items to choose, which is 3.
Plugging in the values:
[tex]\[ C(30, 3) = \frac{30!}{3!(30-3)!} = \frac{30!}{3! \cdot 27!} \][/tex]
We can simplify this expression by canceling out the [tex]\( 27! \)[/tex] in the numerator and denominator:
[tex]\[ C(30, 3) = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} \][/tex]
Now, let's compute the value step-by-step:
1. Multiply the top numbers:
[tex]\[ 30 \times 29 = 870 \][/tex]
[tex]\[ 870 \times 28 = 24,360 \][/tex]
2. Multiply the bottom numbers:
[tex]\[ 3 \times 2 = 6 \][/tex]
[tex]\[ 6 \times 1 = 6 \][/tex]
3. Now, divide the results:
[tex]\[ \frac{24,360}{6} = 4,060 \][/tex]
So, the number of different three-topping pizzas you can get is [tex]\(\boxed{4060}\)[/tex].
Among the given options, the correct answer is:
4,060
This means you can create 4,060 unique pizzas from the 30 different toppings for $11.99 each.
