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Simon is creating a flower arrangement. A client asked him to select 4 of the 7 available flower options to create the arrangement.
How many different groups of 4 flowers can Simon choose?

Simon is creating a flower arrangement. A client asked him to select $4$ of the $7$ available flower options to create the arrangement. $\newline$ How many different groups of $4$ flowers can Simon choose?

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Counting principle

Full solution

Q. Simon is creating a flower arrangement. A client asked him to select

4

of the

7

available flower options to create the arrangement.

\newline

How many different groups of

4

flowers can Simon choose?

Identify problem type: Identify the type of problem. Simon is choosing $4$ flowers out of $7$ without regard to order, which is a combination problem.
Use combination formula: Use the combination formula to calculate the number of different groups. The combination formula is $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, and $\text{“!“}$ denotes factorial.
Plug values into formula: Plug the values into the combination formula. Here, $n = 7$ (total flower options) and $k = 4$ (flowers to choose). $C(7, 4) = \frac{7!}{4!(7-4)!}$
Calculate factorials and simplify: Calculate the factorials and simplify the expression. $\newline$ $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $\newline$ $4! = 4 \times 3 \times 2 \times 1$ $\newline$ $(7-4)! = 3! = 3 \times 2 \times 1$ $\newline$ $C(7, 4) = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1) \times (3 \times 2 \times 1)}$
Cancel out common terms: Cancel out the common terms in the numerator and the denominator. $\newline$ $C(7, 4) = \frac{7 \times 6 \times 5}{3 \times 2 \times 1}$
Perform calculation: Perform the calculation. $\newline$ $C(7, 4) = \frac{7 \times 6 \times 5}{3 \times 2}$ $\newline$ $C(7, 4) = \frac{7 \times 5}{1}$ $\newline$ $C(7, 4) = 35$

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