Combinations are another variation of the fundamental counting principle. **Combinations **come into play when the order of the selected elements from the single set does **not **matter. On the other hand, **Permutations **are like combinations but the order does matter.

- Combinations - Order does NOT matter
- Permutations - Order DOES matter

For example, the ice cream store has a special for a cup of ice cream with five flavors (no toppings), but you cannot choose the same flavor again for any of the five scoops. Choosing chocolate for the first scoop and vanilla for the second scoop is not any different than choosing vanilla first and chocolate second—you still have a scoop of chocolate and a scoop of vanilla in your cup after the first two selections.

**Take a look at the following example involving combinations:**

A debate team has 8 students: Adam, Bridgette, Carl, Donna, Eric, Farrah, George, and Heidi. To split the students into two teams, how many combinations are possible for the first team?

- Option 1 - You choose Adam, then Donna, then Farrah, then George. Team 1 is therefore composed of Adam, Donna, Farrah, and George.
- Option 2 - You choose Donna, then George, then Adam, then Farrah. Team 1 is still composed of the same people--Adam, Donna, Farrah, and George

No matter what order you choose them, your team contains Adam, Donna, Farrah, and George. The order that you chose them in does not matter in this case.

Combinations are used to determine the total number of possible combinations that occur when the order of making selections from a single set does not matter.

The equation for combinations can be expressed in two ways, with and without factorials:

For the equation without factorials, since the final value multiplied in the numerator is determined by (*n* - *r* + 1), the number of values being multiplied together will match the number of choices or selections you are making from the set. Since the final number multiplied in the denominator is (*r* - *r* - 1), the final number is always 1.

Returning to the original problem with ice cream, let’s create combinations for ice cream cups.

There are 792 combinations for making a cup of ice cream with five scoops. The number of combinations is smaller than the same example using permutations because the order does not matter in this case, meaning that we have a smaller number of distinct groups.

- If π = 7 and π = 4, how many combinations are possible?
- If π = 6 and π = 2, how many combinations are possible?
- If π = 11 and π = 4, how many combinations are possible?
- If π = 105 and π = 2, how many combinations are possible?
- A business decides to hang 6 posters in its main conference room. If there are 10 posters from which to choose, how many combinations are possible?
- a. 210
- b. 330
- c. 510
- d. 720

- There are 14 members in a book club, and 4 members will be attending a national book club event. How many combinations of these members could attend the event?
- a. 1,001
- b. 6,006
- c. 12,012
- d. 24,024

- In an experiment, there are 28 possible combinations of two different floras that can be grown. How many types of flora are available for the experiment?
- a. 7
- b. 8
- c. 9
- d. 10

- A zoo has 15 giraffes and its display pen can hold 7 giraffes. How many possible combinations of the giraffes can be put on display?
- a. 6,435
- b. 8,205
- c. 11,625
- d. 15,435

**Answerse to Practice Problems**

- 35
- 15
- 330
- 5,460
- 210
- 1,001
- B
- 6,435

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