Calculating the Probability of Recovering All Math Homework Using Combinatorics

Introduction to the Problem

Cody has a challenging task to complete. He carries seven pages of math homework and three pages of English homework, totaling ten pages. Unfortunately, a gust of wind blows the pages from his hands, and he recovers only seven random pages. Here, we will calculate the probability that Cody recovers all of his math homework using the principles of combinatorics.

Step-by-Step Solution Using Combinatorics

No matter the situation, the principles of combinatorics can help us solve this problem accurately. Let's break it down into several steps.

Step 1: Determine the Total Number of Ways to Choose 7 Pages from 10

The total number of ways to choose 7 pages from a total of 10 pages can be found using the combination formula:

C(n, k) n! / (k! (n-k)!)

In this problem, n 10 (total pages) and k 7 (pages Cody wants to recover).

C(10, 7) C(10, 3) 10! / (3! * 7!)

Simplifying this, we get:

C(10, 7) 10 * 9 * 8 / (3 * 2 * 1) 720 / 6 120

Step 2: Determine the Number of Favorable Outcomes

To find the number of favorable outcomes where Cody recovers all 7 pages of his math homework, we need to ensure that all 7 pages he recovers are math pages. Since there are only 7 math pages and Cody recovers 7 pages, the only favorable outcome is when all 7 pages are math pages. There is only one way for this to happen:

Number of favorable outcomes 1

Step 3: Calculate the Probability

The probability P that Cody recovers all of his math homework is the ratio of the number of favorable outcomes to the total number of outcomes:

P(Cody recovers all math homework) Number of favorable outcomes / Total number of outcomes 1 / 120

Therefore, the probability that Cody recovers all of his math homework is:

1 / 120

Conclusion

Using the principles of combinatorics, we have determined that the probability of Cody recovering all of his math homework is 1 / 120. This approach can be applied to similar problems involving the calculation of probabilities using combinations.

Further Assistance

If you require more help with math homework or need assistance with any other mathematical concepts, feel free to visit my site at My Mathlab Homework.