Cambridge Additional Mathematics

268 Counting and the binomial expansion (Chapter 10)

Example 14 Self Tutor

A committee of 4 is chosen from 7 men and 6 women. How many different committees can be

chosen if:

a there are no restrictions b there must be 2 of each sex

c there must be at least one of each sex?

a

b

c

EXERCISE 10E

1 Determine whether the following are examples of combinations

or permutations:

a making a 3 -digit number using the digits 1 , 2 , 3 , 4 , and 5 at

most once each

b selecting a committee of 3 people from a list of 5

c selecting the chairperson and treasurer from a committee of

8 people

d selecting 2 pieces of fruit to take to school from a bowl of

10 pieces.

2 List the different teams of 4 that can be chosen from a squad of 6 (named A, B, C, D, E, and F).

Check that the formula

¡n

r

¢

=

n!

r!(n¡r)!

gives the total number of teams.

3 How many different teams of 11 can be chosen from a squad of 17?

4 Candidates for an examination are required to answer 5 questions out of 9.

a In how many ways can the questions be chosen if there are no restrictions?

b If question 1 was made compulsory, how many selections would be possible?

5aHow many different committees of 3 can be selected from 13 candidates?

b How many of these committees consist of the president and 2 others?

For combinations,

the order of selection

does not matter.

There are 7+6=13people up for selection and we want any 4 of them.

There are

¡ 13

4

¢

= 715 possible committees.

The 2 men can be chosen out of 7 in

¡ 7

2

¢

ways.

The 2 women can be chosen out of 6 in

¡ 6

2

¢

ways.

) there are

¡ 7

2

¢

£

¡ 6

2

¢

= 315 possible committees.

The total number of committees

=the number with 3 men and 1 woman+the number with 2 men and 2 women

+the number with 1 man and 3 women

=

¡ 7

3

¢

£

¡ 6

1

¢

+

¡ 7

2

¢

£

¡ 6

2

¢

+

¡ 7

1

¢

£

¡ 6

3

¢

= 665

or The total number of committees

=

¡ 13

4

¢

¡the number with all men ¡the number with all women

=

¡ 13

4

¢

¡

¡ 7

4

¢

£

¡ 6

0

¢

¡

¡ 7

0

¢

£

¡ 6

4

¢

= 665

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\268CamAdd_10.cdr Friday, 4 April 2014 2:01:48 PM BRIAN