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