Question

Legislative Committee A legislative committee consists of 5 Democrats and 4 Republicans. A delegation of 3 is to be selected to visit a small Pacific island republic. (a) How many different delegations are possible? (b) How many delegations would have all Democrats? (c) How many delegations would have 2 Democrats and 1 Republican? (d) How many delegations would include at least 1 Republican?

Answer 1

Explanation:

GIVEN: Legislative Committee A legislative committee consists of
`5` Democrats and
`4` Republicans. A delegation of
`3` is to be selected to visit a small Pacific island republic.

TO FIND: (a) How many different delegations are possible? (b) How many delegations would have all Democrats? (c) How many delegations would have
`2` Democrats and
`1` Republican? (d) How many delegations would include at least
`1` Republican.

SOLUTION:

(a)

Total members
`=9`

Total delegation possible
`=^9C_3=(9!)/(6!3!)`

`=(9*8*7)/(6)=84`

(b)

Total democrats
`=5`

Total delegation possible with all democrats
`=^5C_3=(5!)/(3!2!)`

`=10`

(c)

Total democrats
`=4`

Total republicans
`=5`

Total delegations with
`2` democrats and
`1` republican
`=^4C_1*^5C_2=(4!)/(3!1!)*(5!)/(2!3!)`

`=4*10=40`

(d)

Total republicans
`=4`

Total delegation with at least
`1` republican
`=\text{total delegation possible}-\text{total delegation with all democrats}`

putting values from part (a) and (b)

Total delegation with at least
`1` republican
`=84-10=74`