Question

An ice cream store sells 23 flavors of ice cream, determine the number of 4 dip sundaes. how many are possible if order is not considered and no flavor is repeated?

Answer 1

Answer:

8,855

Step-by-step explanation:

The way to solve this problem is by using Combinations.

In Combinations, we can form different collections of k elements from a total of n elements where the order of them does not matter and any member of them is not repeated.

Combinations is expressed mathematically as:

$\\nC_k = \frac{n!}{(n-k)!k!}$ [1]

Where n is the total elements, k is the number of elements selected from n, and n! is n factorial, or, for instance, 3! is 3*2*1 = 6; 4! is 4*3*2*1 = 24.

This formula tells us how to form groups of k members from a total of n elements. These groups of k members have no repeated elements, that is, in the context of this question, no flavor is repeated in any group.

Likewise, different orders of the same members do not matter, or, in other words, if we have two groups of four members flavors (vanilla, chocolate, strawberry, lemon) and (chocolate, vanilla, lemon, strawberry), they are considered the same group since order does not matter in Combinations.

In this way, to determine the number of four dip sundaes (k) from 23 flavors (n) that an ice cream store sells, we need to apply the formula [1], as follows:

$\\23C_4 = \frac{23!}{(23-4)!4!}$

$\\23C_4 = \frac{23!}{19!4!}$

$\\23C_4 = \frac{23*22*21*20*19!}{19!4!}$, since 19!/19! = 1.

$\\23C_4 = \frac{23*22*21*20}{4*3*2*1}$

$\\23C_4 = 8,855$