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

Answer:

Step-by-step 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$=^9C_3=\frac{9!}{6!3!}$

                                         $=\frac{9\times8\times7}{6}=84$

(b)

Total democrats $=5$

Total delegation possible with all democrats $=^5C_3=\frac{5!}{3!2!}$

                                                                          $=10$

(c)

Total democrats $=4$

Total republicans $=5$

Total delegations with $2$ democrats and $1$ republican $=^4C_1\times^5C_2=\frac{4!}{3!1!}\times\frac{5!}{2!3!}$

                                                                                       $=4\times10=40$

(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 $=84-10=74$