A blood bank needs $4$ people to help with a blood drive. $11$ people have volunteered. $\newline$ Find how many different groups of $4$ can be formed from the $11$ volunteers. $\newline$ Answer:

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Grade 8

Experimental probability

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Q. A blood bank needs

4

people to help with a blood drive.

11

people have volunteered.

\newline

Find how many different groups of

4

can be formed from the

11

volunteers.

\newline

Answer:

Identify Problem Type: Identify the type of problem. $\newline$ We need to find the number of combinations of $11$ people taken $4$ at a time, since the order in which the volunteers are chosen does not matter.
Use Combination Formula: Use the combination formula. $\newline$ The number of combinations of $n$ items taken $r$ at a time is given by: $\newline$ $C(n, r) = \frac{n!}{r! \times (n - r)!}$ $\newline$ where $!$ denotes factorial, which is the product of all positive integers up to that number.
Plug in Values: Plug in the values for $n$ and $r$ . $\newline$ In this case, $n = 11$ (total volunteers) and $r = 4$ (number of people needed for the blood drive). $\newline$ $C(11, 4) = \frac{11!}{4! \times (11 - 4)!}$
Calculate and Simplify: Calculate the factorials and simplify. $\newline$ $11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ $4! = 4 \times 3 \times 2 \times 1$ $\newline$ $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ Now we can cancel out the common terms in the numerator and the denominator. $\newline$ $C(11, 4) = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}$
Perform Calculation: Perform the calculation. $\newline$ $C(11, 4) = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}$ $\newline$ $C(11, 4) = \frac{11 \times 10 \times 9 \times 2}{4 \times 3 \times 2}$ $\newline$ $C(11, 4) = \frac{11 \times 10 \times 9}{4 \times 3}$ $\newline$ $C(11, 4) = \frac{11 \times 10 \times 3}{4}$ $\newline$ $C(11, 4) = 11 \times 5 \times 3$ $\newline$ $C(11, 4) = 165$