A blood bank needs $12$ people to help with a blood drive. $18$ people have volunteered. Find how many different groups of $12$ can be formed from the $18$ volunteers. $\newline$ Answer:

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Grade 8

Experimental probability

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Q. A blood bank needs

12

people to help with a blood drive.

18

people have volunteered. Find how many different groups of

12

can be formed from the

18

volunteers.

\newline

Answer:

Identify Problem Type: Identify the type of problem. $\newline$ We need to find the number of combinations of $18$ people taken $12$ at a time. This is a combinatorics problem involving combinations without repetition.
Use Combination Formula: Use the combination formula. $\newline$ The number of ways to choose $k$ people from a group of $n$ people is given by the combination formula: $\newline$ $C(n, k) = \frac{n!}{k!(n - k)!}$ $\newline$ where $n!$ denotes the factorial of $n$ , which is the product of all positive integers up to $n$ .
Apply Formula: Apply the formula to our problem. $\newline$ We have $n = 18$ volunteers and we want to choose $k = 12$ people. So we need to calculate: $\newline$ $C(18, 12) = \frac{18!}{12!(18 - 12)!}$
Simplify Expression: Simplify the expression. $\newline$ $C(18, 12) = \frac{18!}{(12! \cdot 6!)}$ $\newline$ We can cancel out the common factorial terms by expanding the factorials partially: $\newline$ $18! = 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13 \cdot 12!$ $\newline$ Now we can cancel the $12!$ on the numerator and denominator. $\newline$ $C(18, 12) = \frac{(18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13)}{6!}$
Calculate Result: Calculate the result. $\newline$ $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ $\newline$ Now we divide the product of the numbers in the numerator by $720$ : $\newline$ $C(18, 12) = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13}{720}$
Perform Division: Perform the division. $\newline$ $C(18, 12) = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13}{720}$ $\newline$ $C(18, 12) = \frac{18}{6} \times \frac{17}{1} \times \frac{16}{4} \times \frac{15}{5} \times \frac{14}{2} \times \frac{13}{1}$ $\newline$ $C(18, 12) = 3 \times 17 \times 4 \times 3 \times 7 \times 13$ $\newline$ $C(18, 12) = 3 \times 3 \times 4 \times 7 \times 13 \times 17$ $\newline$ $C(18, 12) = 9 \times 4 \times 7 \times 13 \times 17$ $\newline$ $C(18, 12) = 36 \times 7 \times 13 \times 17$ $\newline$ $C(18, 12) = 252 \times 13 \times 17$ $\newline$ $C(18, 12) = 3276 \times 17$ $\newline$ $C(18, 12) = 55692$