How many ways can 3 out of 8 different colored balls be chosen? (2022)

How many ways can 3 out of 8 different colored balls be chosen? (2022)

Step 1: Understand that we want to choose 3 balls from a total of 8 different colored balls.
Step 2: Recognize that this is a combination problem because the order in which we choose the balls does not matter.
Step 3: Use the combination formula, which is written as nCr, where n is the total number of items (balls) and r is the number of items to choose. The formula is nCr = n! / (r! * (n - r)!).
Step 4: In this case, n = 8 (the total number of balls) and r = 3 (the number of balls we want to choose).
Step 5: Plug the values into the formula: 8C3 = 8! / (3! * (8 - 3)!).
Step 6: Calculate 8! (8 factorial), which is 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.
Step 7: Calculate 3! (3 factorial), which is 3 x 2 x 1.
Step 8: Calculate (8 - 3)! = 5!, which is 5 x 4 x 3 x 2 x 1.
Step 9: Substitute these values back into the formula: 8C3 = (8 x 7 x 6) / (3 x 2 x 1).
Step 10: Simplify the calculation: 8 x 7 x 6 = 336 and 3 x 2 x 1 = 6, so 336 / 6 = 56.
Step 11: Conclude that there are 56 different ways to choose 3 balls from 8.

Combinatorics – The problem 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.