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Chapter 6: Problem 30

A bag contains three red marbles, two green ones, one lavender one, two yellows, and two orange marbles. How many possible sets of three marbles are there?

Short Answer

Expert verified
There are 120 possible sets of three marbles.

Step by step solution

01

Calculate the total number of marbles

First, we need to find the total number of marbles in the bag. There are 3 red, 2 green, 1 lavender, 2 yellow, and 2 orange marbles. Total marbles = 3 + 2 + 1 + 2 + 2 = 10 marbles
02

Use the combination formula

Next, we use the combination formula to determine the number of ways to choose 3 marbles out of 10. The combination formula is: \(C(n, k) = \frac{n!}{k!(n-k)!}\) where n is the total number of marbles, k is the number of marbles we want to choose (in this case, 3), and ! denotes a factorial (e.g., 5! = 5 脳 4 脳 3 脳 2 脳 1).
03

Calculate the combinations

Now, apply the combination formula with n = 10 and k = 3: \(C(10, 3) = \frac{10!}{3!(10-3)!}\) \(C(10, 3) =\frac{10!}{3!7!}\)
04

Compute the factorial terms

Compute the factorial terms separately: 10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 3,628,800 3! = 3 脳 2 脳 1 = 6 7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 5,040
05

Calculate the result

Substitute the factorial values back into the combination formula: \(C(10, 3) = \frac{3,628,800}{6 \times 5,040}\) \(C(10, 3) = \frac{3,628,800}{30,240}\) C(10, 3) = 120 There are 120 possible sets of three marbles.

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