Numbers and Operations

3.1 Counting


Many other counting problems use the multiplication principle.


1. Suppose you have 5 shirts, 4 pairs of pants, and 9 ties. How many outfits can be made consisting of a shirt, a pair of pants, and a tie?

For each of the 5 shirts, you can wear 4 pairs of pants, so there are 5  4 = 20 shirt-pants combinations. For each of these 20 shirt-pants combinations, there are 9 ties, so there are 20  9 = 180 shirt-pants-tie combinations.

2. Six very good friends decide they will have lunch together every day. In how many different ways can they line up in the lunch line?

Any one of the 6 could be first in line. For each person who is first, there are 5 who could be second. This means there are 30 (6  5) ways of choosing the first two people. For each of these 30 ways, there are 4 ways of choosing the third person. This makes 120 (30  4) ways of choosing the first 3 people. Continuing in this fashion, there are 6  5  4  3  2  1 = 720 ways these 6 friends can stand in the cafeteria line. (This means that if they all have perfect attendance for 4 years of high school, they could stand in line in a different order every day, because 720 = 4  180.)

3. The math team at East High has 20 members. They want to choose a president, vice president, and treasurer. In how many ways can this be done?

Any one of the 20 members could be president. For each choice, there are 19 who could be vice president. For each of these 380 (20  19) ways of choosing a president and a vice president, there are 18 choices for treasurer. Therefore, there are 20  19  18 = 6840 ways of choosing these three club officers.

4. The student council at West High has 20 members. They want to select a committee of 3 to work with the school administration on policy matters affecting students directly. How many committees of 3 students are possible?

This problem is similar to example 3, so we will start with the fact that if they were electing 3 officers, the student council would be able to do this in 6840 ways. However, it does not matter whether member A is president, B is vice president, and C is treasurer or some other arrangement, as long as all 3 are on the committee. Therefore, we can divide 6840 by the number of ways the 3 students selected could be president, vice president, and treasurer. This latter number is 3  2  1 = 6, so there are 1140 (6840 ÷ 6) committees of 3.


1. M & M plain candies come in six colors: brown, green, orange, red, tan, and yellow. Assume there are at least 3 of each color. If you pick three candies from a bag, how many color possibilities are there?

      (A)  18

      (B)  20

      (C)  120

      (D)  216

      (E)  729

2. A code consists of two letters of the alphabet followed by 5 digits. How many such codes are possible?

      (A)  7

      (B)  10

      (C)  128

      (D)  20,000

      (E)  67,600,000

3. A salad bar has 7 ingredients, excluding the dressing. How many different salads are possible where two salads are different if they don’t include identical ingredients?

      (A)  7

      (B)  14

      (C)  128

      (D)  5,040

      (E)  823,543