1.4 Notes/Examples

Overview of Notes

Class Notes File

We just looked at situations in which order matters. What if order does not matter? Suppose you have a younger brother or sister and your family goes out to a restaurant. There is a children's menu with activities at the restaurant that all the kids get. The owner of the restaurant has decided that each child will receive two different colored crayons to use on their menu. The restaurant happens to carry five colors of crayons: orange, yellow, blue, green, and red.

This is a situation in which the order that the child gets their two color crayons does not matter. If you gave a child a red crayon and then a blue crayon, it would be the same as if you gave the child a blue crayon followed by a red crayon. As with permutations, the first question to ask is "Does Order Matter?". When the order does not matter, you are dealing with a situation that involves combinations.

Example 1

Consider the color crayon problem in the previous situation. Make a list showing all of the different color crayon combinations that might occur. Be organized so as not to repeat any combinations.

[Figure2]

Solution

To be organized, use the letters O, Y, B, G, and R to represent the five colors (Orange, Yellow, Blue, Green, and Red). Alphabetizing the list to insure that we don't skip any combinations gives us BG, BO, BR, BY, GO, GR, GY, OR, OY, RY Notice that while we have BG, we don't have GB as that would be a repeat. It appears that there are 10 combinations possible.

As with the Fundamental Counting Principle, we now must ask the question "How can we find the solution quickly?" Making a list works nice, but it could get a bit messy if the restaurant had 24 colors to choose from instead of 5 because our list would get very long. Out of curiosity, you may have tried

However, when you work this out, you find that this gives us a result of 20 instead of 10. We must modify this formula for situations involving combinations.

Shown below is the formula for finding how many combinations are possible when order does not matter.

[Figure3]

As with the permutation formula, the 'n' stands for the number of objects available and the 'r' stands for the number of objects that will be selected. We will say "n choose r".

Example 2

Consider the color crayon problem once again. Use the formula to find out the number of different color crayon combinations that are possible.

Solution

We would say, "5 choose 2" for this combination. Be very careful that you find the result for the denominator before you divide. Find the nCr command on your calculator and verify that 5C2 is indeed equal to 10.

Example 3

Suppose that there are 12 employees in an office. The boss needs to select 4 of the employees to go on a business trip to California. In how many ways can she do this?

Solution

We first ask whether the order that the employees are selected matters. In this case, the answer is no because either you will be going on the trip or you won't be going. Being the fourth name on the list of people who get to go is just as good as being the first name on the list. We have 12 people to select from and we will be selecting 4 so the combination will be: 12C4

There are 495 possible combination of groups of 4 that might be selected to go on the trip to California.

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