## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 3

## Numbers and Operations

3.1 Counting

### MULTIPLICATION RULE

Many other counting problems use the multiplication principle.

**EXAMPLES**

**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.

**EXERCISES**

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