Question

A legislative committee consists of 9 Democrats and 7 Republicans. A delegation of 4 is to be selected to visit a small republic. a. How many different delegations are​ possible? b. How many delegations would have all​ Democrats? c. How many delegations would have 3 Democrats and 1​ Republican? d. How many delegations would include at least 1​ Republican?

Answer 1

Total people = 9 + 7 = 16

a) Number of different delegations are possible = 16C4 = 16! / (4! * (16 - 4)!) = 1820

b) Number of delegations with all democrats = 9C4 = 9! / (4! * (9 - 4)!) = 126

c) Number of delegations with 3 democrats and 1 republican = 9C3 * 7C1 = 588

d) Number of delegations with at least 1 republican = Number of different delegations are possible - Number of delegations with all democrats

                                             = 1820 - 126

                                             = 1694