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Chapter 6: Problem 34

A bag contains three red marbles, two green ones, one lavender one, two yellows, and two orange marbles. How many sets of three marbles include none of the yellow ones?

Short Answer

Expert verified
There are 56 sets of three marbles that do not include any yellow marbles.

Step by step solution

01

Determine the total number of marbles

The given bag contains: - 3 red marbles - 2 green marbles - 1 lavender marble - 2 yellow marbles - 2 orange marbles This sums up to \(10\) marbles in total.
02

Calculate the total number of sets of 3 marbles

Use combinations (denoted as C(n, k)) to find the total number of ways to choose 3 marbles out of 10: \(C(10, 3) = \frac{10!}{3! (10-3)!} = \frac{10!}{3! 7!} = 120\) There are 120 possible sets of three marbles.
03

Calculate the number of sets including yellow marbles

There are two yellow marbles in the bag. We want to find the number of sets of three marbles that include at least one yellow marble. There are two possibilities: Case 1: Exactly one yellow marble in the 3-marble set: \(C(2, 1)\ 脳 C(8, 2) = 2 脳 28 = 56\) Case 2: Exactly two yellow marbles in the 3-marble set: \(C(2, 2)\ 脳 C(8, 1) = 1 脳 8 = 8\) Now, add the possibilities together to find the total number of sets including yellow marbles: \(56 + 8 = 64\) There are 64 sets of three marbles which include at least one yellow marble.
04

Calculate the number of sets without yellow marbles

To find the number of sets of three marbles without yellow marbles, subtract the number of sets including yellow marbles (found in Step 3) from the total number of sets of three marbles (found in Step 2): \(120 - 64 = 56\) There are 56 sets of three marbles that do not include any yellow marbles.

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