How many ways can you select 3 out of 7 different colored shirts?

How many ways can you select 3 out of 7 different colored shirts?

Step 1: Understand that you want to choose 3 shirts from a total of 7 shirts.
Step 2: Recognize that the order in which you choose the shirts does not matter. This means you will use combinations, not permutations.
Step 3: Use the combination formula C(n, r) = n! / (r! * (n - r)!), where n is the total number of items (shirts) and r is the number of items to choose.
Step 4: In this case, n = 7 (the total shirts) and r = 3 (the shirts you want to select).
Step 5: Plug the numbers into the formula: C(7, 3) = 7! / (3! * (7 - 3)!)
Step 6: Calculate 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1, which is 5040.
Step 7: Calculate 3! = 3 × 2 × 1, which is 6.
Step 8: Calculate (7 - 3)! = 4! = 4 × 3 × 2 × 1, which is 24.
Step 9: Now substitute back into the formula: C(7, 3) = 5040 / (6 * 24).
Step 10: Calculate 6 * 24 = 144.
Step 11: Finally, divide 5040 by 144 to get 35.
Step 12: Therefore, there are 35 different ways to select 3 shirts from 7.

Combinatorics – The question tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset from a larger set without regard to the order of selection.