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Amari has 8 paint colors available, and he wants to mix 3 of them to create a new color.
How many different sets of 3 colors can Amari choose?

◻

Amari has $8$ paint colors available, and he wants to mix $3$ of them to create a new color. $\newline$ How many different sets of $3$ colors can Amari choose? $\newline$ $\square$

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Counting principle

Full solution

Q. Amari has

8

paint colors available, and he wants to mix

3

of them to create a new color.

\newline

How many different sets of

3

colors can Amari choose?

\newline

\square

Understand the problem: Understand the problem. $\newline$ Amari has $8$ different paint colors and wants to choose $3$ of them to mix. We need to find the number of different combinations of $3$ colors he can choose from the $8$ available colors.
Recognize order doesn't matter: Recognize that the order in which the colors are chosen does not matter. $\newline$ This is a combination problem, not a permutation, because the order of the colors in the mix does not change the outcome.
Use combination formula: Use the combination formula to calculate the number of different sets. $\newline$ The number of ways to choose $3$ items from $8$ is given by the combination formula: $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, ＂ $!$ ＂ denotes factorial, and $C(n, k)$ is the number of combinations.
Plug in values: Plug in the values into the combination formula. $\newline$ Here, $n = 8$ (total colors) and $k = 3$ (colors to choose). So, $C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}$ .
Calculate factorials: Calculate the factorials and simplify the expression. $\newline$ $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ $3! = 3 \times 2 \times 1$ $\newline$ $5! = 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ Now, $C(8, 3) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (5 \times 4 \times 3 \times 2 \times 1)}$
Cancel common terms: Cancel out the common terms in the numerator and the denominator. $\newline$ $C(8, 3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}$
Perform calculation: Perform the calculation. $\newline$ $C(8, 3) = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{8 \times 7 \times 6}{6} = 8 \times 7 = 56$
Conclude with final answer: Conclude with the final answer. $\newline$ Amari can choose $56$ different sets of $3$ colors from the $8$ available colors.

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