How many possible combinations of 6 items are possible?

Answer 1

Situation 1: Organizing 6 items

Let's say you are placing 6 books on a shelf, how many possible combinations of 6 items are possible?

You could pick any of the six books for the first spot. For the second spot you have five books to choose from. You have four options for the third spot, three for the fourth spot, two options for the fifth spot, and the last book goes in the sixth spot.

Mathematically speaking, that would be 6 x 5 x 4 x 3 x 2 x 1 = 720 possible combinations.

Situation 2: Choosing # of items

You have finished sorting the books and you are hungry. You decide to go to McDonalds and buy a hamburger. You can pick 6 toppings, cheese, lettuce, tomatoes, mustard, ketchup, and onions. If you are only allowed two toppings, how many possible topping combinations are there? If you could choose three toppings? Four toppings? Let's break it down.

If you could have one topping, you have 6combinations.

If you could have two toppings, you have 30 combinations. Using the same logic as with the books, there are six toppings for the first spot and five toppings for the second spot. That is 6 x 5 = 30 options.

If you could have three toppings, you have 120combinations. Six toppings for the first spot, five for the second, and three for the third topping. That is 6 x 5 x 4 = 120combinations.

If you could have four toppings, you have 6 x 5 x 4 x 3 = 360 combinations.

If you could have four toppings, you have 360 x 2 = 720combinations.