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. How many unique ways are there to arrange 4 of the 6 swimmers?

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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. The number of ways to choose 4 swimmers from 6 is calculated using the combination formula: Number of combinations = 6C4 = 6! / (4! * (6-4)!)Calculate Combinations: Now we calculate the factorial values needed for the combination formula. 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 4! = 4 × 3 × 2 × 1 = 24 (6-4)! = 2! = 2 × 1 = 2Calculate Factorials: We substitute the factorial values into the combination formula to find the number of ways to choose 4 swimmers. 6C4 = 720 / (24 * 2)Substitute and Divide: Perform the division to find the number of combinations. 6C4 = 720 / 48 6C4 = 15 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. The number of ways to arrange 4 swimmers is calculated using the permutation formula for n objects taken r at a time: Number of permutations = P(n, r) = n! / (n-r)! 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. Number of permutations = P(4, 4) = 4! / (4-4)!Factorial Calculation: Now we calculate the factorial values needed for the permutation formula. 4! = 4 × 3 × 2 × 1 = 24 (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. P(4, 4) = 24 / 1 P(4, 4) = 24 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. Total unique arrangements = Number of combinations × Number of permutations Total unique arrangements = 15 × 24Perform Final Multiplication: Perform the multiplication to find the total number of unique arrangements. Total unique arrangements = 15 × 24 Total unique arrangements = 360

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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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?