A blood bank needs $12$ people to help with a blood drive. $19$ people have volunteered. Find how many different groups of $12$ can be formed from the $19$ volunteers. $\newline$ Answer:

Full solution

Q. A blood bank needs

12

people to help with a blood drive.

19

people have volunteered. Find how many different groups of

12

can be formed from the

19

volunteers.

\newline

Answer:

Identify Concept: Identify the mathematical concept needed to solve the problem. $\newline$ To find the number of different groups of $12$ that can be formed from $19$ volunteers, we need to use the concept of combinations. The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items, $k$ is the number of items to choose, and “ $!$ ” denotes factorial.
Apply Formula: Apply the formula for combinations to the given numbers. $\newline$ We have $19$ volunteers in total ( $n = 19$ ) and we want to form groups of $12$ ( $k = 12$ ). So we need to calculate $C(19, 12) = \frac{19!}{12!(19-12)!}$ .
Simplify Expression: Simplify the expression by calculating the factorials and the division. $C(19, 12) = \frac{19!}{12! \times 7!} = \frac{19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$ since the factorials from $1$ to $12$ cancel out with part of the $19!$ factorial.
Perform Calculations: Perform the calculations. $\newline$ $C(19, 12) = \frac{19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{50388}{5040} = 10$