We want to order a burrito and we have a choice of 2 main ingredients and 3 toppings. We are interested in finding how many different burritos we can order. Therefore, we have to consider two events.
A: choice of 2 main ingredients
B: choice of 3 toppings
First, let's find how many choices of 2 main ingredients we have. We want to choose 2 main ingredients out of 6 possible. Notice the order in which we choose the main ingredients is not important, since we are interested only in different kinds of them. Let's recall the formula for the number of combinations of n objects taken r at a time, where r ≤ n.
_nC_r = n!/(n-r)! * r!In our case we can choose from 6 different main ingredients, so n = 6. Out of them we pick 2 main ingredients, therefore r = 2. Let's substitute these values into the formula.
We found that there are 15 different combinations of two main ingredients we can choose. Similarly, let's find the number of combinations of 3 toppings out of 8 possible toppings. We need to evaluate _8C_3.
We found there are 56 different combinations of toppings. Notice the choice of ingredients is independent from the choice of the toppings. That enables us to use the Fundamental Counting Principle, since this principle is used to find the number of possible outcomes for a combination of independent events.
Fundamental Counting Principle
If an Event A has n possible outcomes and an Event B has m possible outcomes, then the total number of different outcomes for A and B combined is n * m.
Therefore, to find how many different burritos we can order we need to multiply the number of combinations of choices for main ingredients and toppings.
Let's substitute the obtained values and evaluate the number of possible different burritos.
15 * 56 = 840
We can order 840 different burritos.
