Counting

1

In how many different ways can five persons be seated on a bench?

$$5!=120$$

2

How many three-digit odd numbers can be formed with the digits 1, 2, . . . , 9 if no digit is repeated in any number?

Odd numbers must end with 1, 3, 5, 7, 9

1. Choose odd number for units place
2. Choose remaining 8 for hundreds place
3. Choose remaining 7 for tens place

$$5\times 8\times 7=280$$

3

In how many ways can three boys and three girls be seated in a row if boys and girls alternate?

1. BGBGBG
2. GBGBGB
3. The boys/girls in the above order can be different persons

$$3!\times 3!\times 2=72$$

4

In how many ways can two letters be mailed if five letter boxes are available?

$$5\times 5=25$$

5

In how many ways can 10 boys take positions in a straight line if two particular boys must not stand side by side?

<> B <> B <> B <> B <> B <> B <> B <> B <>

1. Arrange 8 boys
2. Permute 2 boys in 9 slots

$$8!\times 9\times 8=2903040$$

1. Arrange 10 boys
2. Subtract the ways to put the 2 boys together (group 2 boys, times the way to permute them)

$$10!-9!\times 2!$$

6

In how many ways can the offices of chairman, vice-chairman, secretary, and treasurer be filled from a committee of seven?

$$P_4^7=840$$

7

How many three-digit numbers greater than 300 can be formed with the digits 1, 2, . . . , 6 if no digit is repeated in any number?

$$4\times 5\times 4=80$$

8

A bag contains nine balls numbered 1, 2, . . . , 9. In how many ways can two balls be drawn so that (a) both are odd? (b) their sum is odd?

Note that the question is asking for combination, not permutation

a

$$\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$$

b

Sum is odd if odd + even (5 odd, 4 even)

$$5\times 4=20$$

9

Two dice can be tossed in 36 ways. In how many of these is the sum equal to (a) 4; (b) 7; (c) 11?

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

a

3 ways

b

6 ways

c

2 ways

10

Four delegates are to be chosen from eight members of a club. (a) How many choices are possible? (b) How many contain member A? (c) How many contain A or B but not both?

a

$$\left(\genfrac{}{}{0ex}{}{8}{4}\right)=70$$

b

$$\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$$

c

1. All choices
2. Subtract choices that contains both A and B
3. Subtract choices that do not contain A or B

$$\left(\genfrac{}{}{0ex}{}{8}{4}\right)-\left(\genfrac{}{}{0ex}{}{6}{2}\right)-\left(\genfrac{}{}{0ex}{}{6}{4}\right)=40$$

1. Choose remaining 3 (no A or B)
2. Choose A or B

$$\left(\genfrac{}{}{0ex}{}{6}{3}\right)\times 2=40$$