Lesson 118: Solving Problems Involving Combinations

New Concept

A combination is a grouping of items where order does not matter.

What’s next

Next, you’ll use the combination formula to calculate possible outcomes and determine probabilities in various scenarios.

Property

A combination is a grouping of items where order does not matter.

Explanation

Think of picking pizza toppings. Choosing mushrooms then pepperoni is the same as pepperoni then mushrooms—you get the same delicious pizza! A combination is just the final group, where the selection order is irrelevant. It’s all about the final set of items you end up with, not the specific path you took to pick them.

Examples

Choosing 3 friends from a group of 5 to go to the movies is a combination because the group {Ann, Bob, Cid} is the same as {Bob, Cid, Ann}.
Selecting 2 side dishes from 6 options at a restaurant is a combination; mac & cheese and fries is the same as fries and mac & cheese.
Picking 4 markers from a box of 8 to color a map is a combination since the handful of markers is the same regardless of picking order.

When choosing a group, does the order of selection change the outcome? Let's explore why sometimes it doesn't. This example highlights the key idea of how combinations relate to permutations.

Example Problem: A club has 5 members. How many distinct committees of 3 members can be formed?

1. First, let's find the number of ways to arrange 3 members out of 5. This is a permutation.

There are 60 ordered arrangements.

2. However, for a committee, the order doesn't matter. A committee with members {A, B, C} is the same regardless of who was picked first.
3. For any group of 3 members, there are $3!$ ways to arrange them ($3 \cdot 2 \cdot 1 = 6$). These 6 arrangements all represent the same single committee.
4. To find the number of unique committees (combinations), we must divide the total permutations by the number of arrangements for each group.

5. Calculating this gives us the answer:

There are 10 possible committees.

Property

The number of combinations of $n$ items taken $r$ at a time is

Explanation

This awesome formula lets you count combinations without listing every single one! It cleverly starts with all the permutations (ordered groups) and then divides by the number of ways to arrange the chosen items ($r!$). This removes all the duplicates, since with combinations, order doesn't matter. It’s a powerful shortcut to find just the unique groups.

Examples

To find the combinations of choosing 2 side dishes from 6: $_6C_2 = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = 15$.
To find the combinations of choosing 3 test questions from 4: $_4C_3 = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = 4$.
How many ways can a customer choose 12 fruit types from 16 options? $_{16}C_{12} = \frac{16!}{12!4!} = 1820$.

Let's use our powerful new formula to tackle a bigger problem without listing all the possibilities. This example will show you how to apply the combination formula.

Example Problem: An ice cream shop offers 15 different toppings. How many ways can you choose 12 toppings for your sundae?

1. We need to find the number of combinations of 15 items taken 12 at a time. The formula is:

2. Substitute $n=15$ for the total number of toppings and $r=12$ for the number of toppings to choose.

3. Simplify the expression inside the parentheses:

4. To make this easier to calculate, we expand $15!$ until we reach $12!$ so we can cancel the terms.

5. Now, cancel out the $12!$ from the numerator and the denominator:

6. Finally, calculate the result by expanding $3!$ in the denominator.

There are 455 ways to choose 12 toppings from 15.

Property

The theoretical probability of an event is

Explanation

Want to know the chances of a specific group being chosen? First, use the combination formula to find the total number of possible groups—this becomes your denominator. Since you usually want just one specific group (like your three favorite dogs), the number of favorable outcomes is 1. Put that 1 on top, and you've found the probability!

Examples

An animal shelter has 20 dogs. What's the probability that your 3 favorites, {Jumbo, Fluffy, Max}, are chosen? Total combinations: $_{20}C_3 = 1140$. Probability: $\frac{1}{1140}$.
A cook with 18 ingredients picks 4 for a soup. The probability he picks {beans, corn, rice, and carrots} is $\frac{1}{_{18}C_4} = \frac{1}{3060}$.