Question

diana baniak, searching for an ecology class, knows that it must be in one of ten classrooms. since the professor does not allow people to enter after the class has begun, and there is very little time left, she decides to try just four of the rooms at random. how many of the possible different selections of four rooms will succeed in locating the class?
there are successful selections.

Explanation:

Step1: Identify the key - concept

We use combination formula. The class is in one room. We want to find number of combinations with that room included.

Step2: Calculate combinations

The total number of ways to choose 4 rooms out of 10 is given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 10$ and $k=4$. But we want to find the number of successful combinations. A successful combination is one that contains the room with the class. So we fix one room (the room with the class) and then choose the remaining $4-1 = 3$ rooms from the remaining $10 - 1=9$ rooms.
Using the combination formula $C(n,k)$ with $n = 9$ and $k = 3$, we have $C(9,3)=\frac{9!}{3!(9 - 3)!}=\frac{9!}{3!6!}$.
$9! = 9\times8\times7\times6!$, so $C(9,3)=\frac{9\times8\times7\times6!}{3!6!}$.
Since $3!=3\times2\times1 = 6$, then $C(9,3)=\frac{9\times8\times7}{6}=84$.

Answer:

84