In how many ways can 3 different colored balls be chosen from a set of 7?

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Question: In how many ways can 3 different colored balls be chosen from a set of 7?

Options:

Correct Answer: 35

Solution:

The number of ways to choose 3 from 7 is given by C(7,3) = 7!/(3!4!) = 35.

In how many ways can 3 different colored balls be chosen from a set of 7?

In how many ways can 3 different colored balls be chosen from a set of 7?

Step 1: Understand that we have 7 different colored balls.
Step 2: We want to choose 3 balls from these 7.
Step 3: The formula to find the number of ways to choose 'r' items from 'n' items is C(n, r) = n! / (r! * (n - r)!).
Step 4: In our case, n = 7 (the total number of balls) and r = 3 (the number of balls we want to choose).
Step 5: Plug the values into the formula: C(7, 3) = 7! / (3! * (7 - 3)!).
Step 6: Simplify the formula: C(7, 3) = 7! / (3! * 4!).
Step 7: Calculate 7! (which is 7 x 6 x 5 x 4 x 3 x 2 x 1) and 3! (which is 3 x 2 x 1) and 4! (which is 4 x 3 x 2 x 1).
Step 8: Substitute the values: C(7, 3) = 5040 / (6 * 24).
Step 9: Calculate the denominator: 6 * 24 = 144.
Step 10: Now divide: 5040 / 144 = 35.
Step 11: Therefore, there are 35 different ways to choose 3 balls 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.