Combinations and Permuations – Counting Methods

I wrote a couple of posts about combinations and permutations. These posts included definitions and examples of working with both permutations and combinations, but I know reading them is not always the best way for students to learn, so here is a video lesson of the two counting methods: combinations and permutations.



Related to the permutation is the combination. Both permutations and combinations are counting methods, but there is a distinct difference. Remember, when using a permutation the order is important. That is to say if you are to order a pepperoni, onion and mushroom pizza or you ordered a mushroom, pepperoni and onion pizza. Most people would say that is the same pizza, but if one thinks as the previously mentioned pizzas are different, then the person would have to use a permutation to count the situation.

My home made pizza. I usually make two!

Speaking of pizza, here is my home made pizza. I usually make two!

Counting with combinations means that order is not important and just rearranging the items does not count as a different combination. As can be seen in the formula. There is an extra factor in the denominator. I say extra factor in comparison to the permutation formula.

Combination Formula

Combination Formula

The additional factor of r! in the denominator divides out the counting different arrangements of the same objects as separate. Thus combinations usually give a smaller number of arrangements. Since I like pizza so much, here is an example that involves pizza.

Example – Combinations

Nate and wife like to go out to have pizza every Friday night. They to their favorite pizza joint Marcello’s located downtown. Marcello’s offers fifteen different toppings and has a special deal on 5 topping pizzas. The special is a large thick crust pizza loaded with 5 toppings. Regular crust could not hold all the pizza toppings. All this pizza goodness for $12.99. Anyway, Nate and Janet started talking about how many different pizzas there were to choose from at Marcello’s. Help them figure it out.

Since the order of the toppings on the pizza does not matter or make it a different pizza, a combination should be used. There are 15 toppings to choose from and they are allow to have 5. So, one must take 15 toppings taken 5 at a time.

15 pizza toppings taken 5 at a time.

15 pizza toppings taken 5 at a time.

Below is my best attempt at a quick photo shop job to work out the above combination with all of the work shown.

15combo5work-pilarskiIf this has helped you in anyway, leave a comment. If you still have a question you can comment your questions and I will be happy to answer you.

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