Question

select the correct answer

an ice cream shop has 15 different toppings for sundaes, and it is running a special for 3 free toppings. how many 3 - topping sundaes can be made, assuming all 3 toppings chosen are different?

a. 182
b. 455
c. 1,365
d. 2,730

Explanation:

Step1: Identify combination formula

We use combinations since the order of toppings does not matter. The formula for combinations is:
$$C(n,k)=\frac{n!}{k!(n-k)!}$$
where $n=15$ (total toppings) and $k=3$ (toppings chosen).

Step2: Substitute values into formula

$$C(15,3)=\frac{15!}{3!(15-3)!}=\frac{15!}{3! \times 12!}$$

Step3: Simplify the expression

Cancel out $12!$ from numerator and denominator:
$$\frac{15 \times 14 \times 13}{3 \times 2 \times 1}$$

Step4: Calculate the result

$$\frac{15 \times 14 \times 13}{6}=\frac{2730}{6}=455$$

Answer:

B. 455