A swimming coach needs to choose a team for a relay race. The coach must select $4$ of $6$ available swimmers and put them in a strategic sequence. $\newline$ How many unique ways are there to arrange $4$ of the $6$ swimmers?

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Q. A swimming coach needs to choose a team for a relay race. The coach must select

4

6

available swimmers and put them in a strategic sequence.

\newline

How many unique ways are there to arrange

4

of the

6

swimmers?

Select Swimmers: First, we need to select $4$ swimmers out of the $6$ available. This is a combination problem because the order in which we select the swimmers does not matter at this point. $\newline$ The number of ways to choose $4$ swimmers from $6$ is calculated using the combination formula: $\newline$ Number of combinations = $6C4 = \frac{6!}{4! \times (6-4)!}$
Calculate Combinations: Now we calculate the factorial values needed for the combination formula. $\newline$ $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ $\newline$ $4! = 4 \times 3 \times 2 \times 1 = 24$ $\newline$ $(6-4)! = 2! = 2 \times 1 = 2$
Calculate Factorials: We substitute the factorial values into the combination formula to find the number of ways to choose $4$ swimmers. $\newline$ $6C4 = \frac{720}{(24 \times 2)}$
Substitute and Divide: Perform the division to find the number of combinations. $\newline$ $6C4 = \frac{720}{48}$ $\newline$ $6C4 = 15$ $\newline$ This means there are $15$ ways to choose $4$ swimmers from $6$ .
Arrange Swimmers: Next, we need to arrange the $4$ selected swimmers in a strategic sequence. This is a permutation problem because the order in which we arrange the swimmers matters. $\newline$ The number of ways to arrange $4$ swimmers is calculated using the permutation formula for $n$ objects taken $r$ at a time: $\newline$ Number of permutations = $P(n, r) = \frac{n!}{(n-r)!}$ $\newline$ Since we have already chosen $4$ swimmers, we need to arrange these $4$ , so $n = r = 4$ .
Calculate Permutations: We calculate the permutation for $4$ swimmers. $\newline$ Number of permutations = $P(4, 4) = \frac{4!}{(4-4)!}$
Factorial Calculation: Now we calculate the factorial values needed for the permutation formula. $\newline$ $4! = 4 \times 3 \times 2 \times 1 = 24$ $\newline$ $(4-4)! = 0! = 1$ (By definition, $0!$ is equal to $1$ )
Substitute and Find: We substitute the factorial values into the permutation formula to find the number of ways to arrange the swimmers. $\newline$ $P(4, 4) = \frac{24}{1}$ $\newline$ $P(4, 4) = 24$ $\newline$ This means there are $24$ ways to arrange $4$ swimmers.
Multiply for Total: Finally, we multiply the number of combinations by the number of permutations to find the total number of unique ways to arrange $4$ of the $6$ swimmers. $\newline$ Total unique arrangements = Number of combinations $\times$ Number of permutations $\newline$ Total unique arrangements = $15 \times 24$
Perform Final Multiplication: Perform the multiplication to find the total number of unique arrangements. $\newline$ Total unique arrangements = $15 \times 24$ $\newline$ Total unique arrangements = $360$