**Gabrielle **

**B8:** Determine probabilities using permutations and combinations

**Explanation:**

**Probability** is a way to calculate the likeliness that something will happen. It can be calculated using the formula __# of favorable outcomes__

**# of possible outcomes**

Permutations and combinations are the number of ways something can happen. You can use them to calculate either the number of favorable outcomes, or the number of possible outcomes. You must put them into the formula for probability to determine probability. Permutations and combinations DO NOT determine probability.

A **permutation** is a situation in which order does matter. It is represented as **nPr: n being the # of things total, and r being the number you must choose, rearrange etc.** The formula to calculate permutations is: **nPr =**__ n! __

** (n-r)!**

ex. In how many ways could the 1st, 2nd and 3rd place winners be picked for a contest entered by 6 people?

-**This is a permutation because it makes a difference if you are chosen first, or third.**

-** 6 P 3** **=**__ 6!__

** (6-3)!**

** 6 P 3 = 120**

**There are 120 ways that that 1st 2nd and third place winners could be chosen.**

A **combination** is a situation in which order does not matter. It is represented as** n C r: n being the # of things total and **

**r being the number you must choose, rearrange etc.** The formula used to calculate combinations is **nCr=**__ n! __

** r!(n-r)!**

ex. You have to randomly choose 3 out of 10 children to take on a camping trip. How many different ways can you choose them?

-**This is a combination because the choice is random, an regardless of when a child is selecte they get to go on the trip.**

**-20 C 3 = **__ 20! __

** 3!( 20-3)!**

** -20 C 3 = 1140**

** There are 1140 ways that you can choose the children.**

Now that you have gained a basic understanding of probability, combinations, and permutations, this example problem should be no trouble at all!

Try using these steps to ensure you do the problem correctly:

**1. Read the question over carefully, making note of how many objects (people, marbles, golf balls etc.) there are to choose from in total.**

**2. Make note of how many objects there are in separate catagories (colors of golf balls, genders of people etc.)**

**3. Determine if you need to use combinations or permutations. Does order matter?**

**4. Place the combinations or permutations into the formula for probability.**

**Now you try it!**

**Example Problem:**

Mr. Sea is having tryouts for the Mathaletes team for the 2007-2008 school year. Although it would be nice to have everyone participate, there are only 6 spaces on the team, and there are 47 students trying out. 33 of the students trying out are boys, and 14 are girls. Once you have made the team you get the same amount of time solving problems in competition as everyone else. It's making it that's the hard part! With all the talented students at Jay-Elle Milsley High School Mr. Sea is going to have a tough decision to make...

a. What is the probability that Mr. Sea will choose 6 girls for the team?

b. What is the probability no girls will make the team?

**Solution:**

**a. **__14 C 6 __ = __ 3003 __

**47 C 6 10737573 The probability that Mr. Sea will choose 6 girls is 3003/10737573**

The bottom shows all of the possible outcomes. Out of 47 people, 6 must be chosen.

The top shows the favorable outcomes. Out of 14 girls, 6 must be chosen.

This is a combination because there is no ranking on the team. If you make it you are equal with all other team members.

b.** **__33 C 6__ = __107568__ or about 1/10

**47 C 6 10737573 The probability that Mr. Sea will choose all boys is 107568/10737573**

The bottom shows all possible outcomes. Out of 47 people 6 must be chosen.

The top shows all favorable outcomes. Out of 33 boys 6 must be chosen.

**Good luck with you endeavors in calculating probability!**

## Comments (1)

## Anonymous said

at 7:42 pm on Jun 14, 2007

Very nice work!

You don't have permission to comment on this page.