Question

15. A legislative committee consists of 9 Democrats and 6 Republicans. A delegation of 4 is to be selected to visit a small

island republic. Complete parts (a) through (d) below.
(a) How many different delegations are possible?
The 4 delegates can be selected
different ways.
(b) How many delegations would have all Democrats?
The 4 delegates can be selected
different ways if all 4 are Democrats.
(c) How many delegations would have 3 Democrats and 1 Republican?
The delegates can be selected
different ways if 3 are Democrats and 1 is a Republican.
(d) How many delegations would include at least 1 Republican?
The 4 delegates can be selected
different ways if at least 1 is a republican

Answer 1

a) 1365 different ways.

b) 126 different ways.

c) 504 different ways.

d) 1239 different ways.

Explanation:

The order in which the delegates are chosen is not important, which means that we use the combinations formula to solve this question.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

(a) How many different delegations are possible?

4 delegates from a set of 15. So

`C_(15,4) = (15!)/(4!11!) = 1365`

1365 different ways.

(b) How many delegations would have all Democrats?

4 delegates from a set of 9. So

`C_(9,4) = (9!)/(4!5!) = 126`

126 different ways.

(c) How many delegations would have 3 Democrats and 1 Republican?

3 from a set of 9 and 1 from a set of 6. So

`C_(9,3)*C_(6,1) = (9!)/(3!6!)*(6!)/(1!5!) = 84*6 = 504`

504 different ways.

(d) How many delegations would include at least 1 Republican?

Subtract the number of delegations with only democrats(126) from the total(1365). So

1365 - 126 = 1239

1239 different ways.