First, Tosin wants to know how many different sandwiches she can create if she doesn't re-use ingredients. The more variety, the better, right?
Assume that Tosin wants to use 3 different ingredients in a sandwich, and she has 6 ingredients (ham, turkey, cheese, mayonnaise, lettuce, and tomato) available to her. The bread is the same for all sandwiches. How can Tosin determine the number of unique sandwiches she can create?
We can figure out the answer by modifying the equation we would use if we were trying to figure out the number of permutations Tosin could create.
Remember that, for a permutation without repetition, you use the factorial function.
\(\mathsf{ P(n,r) = \frac{n!}{(n-r)!} }\)
However, Tosin is looking for combinations, not permutations. How can we modify that formula to get the correct answer? Well, we know that there are r positions that need to be filled. We also know that, given r positions, there are r! possible ways to rearrange those positions (based on the Fundamental Principle of Counting). We'll use that fact to derive the formula we need.
Click through the slides below to learn how the formula is derived. When you have a good idea of how the formula works, try answering the question below.
|
Calculate the Number of Permutations We start by using the equation to calculate the number of permutations. The equation for calculating the number of permutations, when repetition isn't allowed, is \(\mathsf{ P(n,r) = \frac{n!}{(n-r)!} }\). Order Doesn't Matter In permutations, the order matters. In combinations, the order doesn't matter. Therefore, we need to factor out any repetition taken into account by the permutation equation. Because there are r positions to fill, we know that there are r! possible orientations of those positions. Therefore, we know that for every combination, there are r! orientations of that combination. Finish the Equation Because every combination has r! orientations, those different orientations would count as different permutations. Therefore, we need to divide by r! in order to factor out the extra counts given by the permutation equation. Therefore, the final equation for calculating the number of combinations is the number of permutations divided by r!. |
 |
Combinations without Repetition
C(r, n) = \(\mathsf{ \frac{P(r,n)}{r!} = \frac{n!}{r!(n-r)!} }\)
Do you have a firm grasp of the equation? Try answering the question below.
Question
If Tosin has turkey, cheese, ham, and lettuce available to her and she wants to make a sandwich using 3 of these ingredients (without repeating any), how many possible sandwiches can she make? The bread for all the sandwiches is the same.
