/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 A bag contains three red marbles... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 31

A bag contains three red marbles, two green ones, one lavender one, two yellows, and two orange marbles. How many sets of four marbles include all the red ones?

Short Answer

Expert verified
There are 7 possible sets of four marbles that include all the red ones.

Step by step solution

01

Identify the number of marbles in each color

We have a total of 10 marbles with the following distribution: - 3 red marbles - 2 green marbles - 1 lavender marble - 2 yellow marbles - 2 orange marbles Since we want to include all the red marbles, we already have 3 marbles in a set, and we need to select one more marble from the remaining 7 (10 - 3) marbles.
02

Calculate the combinations of selecting one marble from the remaining 7

To choose one extra marble, we will use the combination formula, which is defined as: \[ C(n,k) = \frac{n!}{k!(n-k)!} \] where \(n\) is the total number of marbles we want to choose from (7 marbles), \(k\) is the number of marbles we want to select (1 marble in this case), and \(C(n, k)\) is the number of different combinations we can get. Applying the formula: \[ C(7,1) = \frac{7!}{1!(7-1)!} \]
03

Calculate the factorials and find the answer

Now, we need to calculate the factorials: - \(7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5040\) - \(1! = 1\) - \((7-1)! = 6! = 720\) Plug the values into the formula: \[ C(7,1) = \frac{5040}{1 \cdot 720} \] \[ C(7,1) = 7 \] So, there are 7 possible sets of four marbles that include all the red ones.

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