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Luis is packing a bag for vacation. He has 9 unique shirts, but he can only fit 5 in his bag.
How many different groups of 5 shirts can he take?

Luis is packing a bag for vacation. He has $9$ unique shirts, but he can only fit $5$ in his bag. $\newline$ How many different groups of $5$ shirts can he take?

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Counting principle

Full solution

Q. Luis is packing a bag for vacation. He has

9

unique shirts, but he can only fit

5

in his bag.

\newline

How many different groups of

5

shirts can he take?

Identify Problem Type: Identify the type of problem. $\newline$ We need to determine the number of combinations of $5$ shirts that can be chosen from a set of $9$ unique shirts. This is a combinatorics problem, specifically a combination problem where order does not matter.
Use Combination Formula: Use the combination formula to calculate the number of different groups. $\newline$ 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. $\newline$ For this problem, $n = 9$ (total unique shirts) and $k = 5$ (shirts to fit in the bag).
Calculate Factorial of $n$ : Calculate the factorial of $n$ , which is $9!$ . $9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880$
Calculate Factorial of $k$ : Calculate the factorial of $k$ , which is $5!$ . $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Calculate Factorial of $(n - k)$ : Calculate the factorial of $(n - k)$ , which is $(9 - 5)!$ . $(9 - 5)! = 4! = 4 \times 3 \times 2 \times 1 = 24$
Substitute into Formula: Substitute the factorials into the combination formula to find the number of combinations. $C(9, 5) = \frac{9!}{5!(9 - 5)!} = \frac{362,880}{120 \times 24}$
Perform Calculations: Perform the calculations. $\newline$ $C(9, 5) = \frac{362,880}{120 \times 24} = \frac{362,880}{2,880} = 126$
Verify Calculations: Verify the calculations to ensure there are no math errors. $\newline$ Rechecking the calculations: $362,880 / 2,880 = 126$ . There are no math errors in the calculations.

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