Question

Consider a political discussion group consisting of 15 Democrats, 11 Republicans, and 8 Independents. 1. Find the number of three person subcommittees that can be made by choosing any republican, any democrat, and any independent. 2. Find the total number of possible 5 person subcommittees that can be chosen if the members can be any of the 34 people in the group. 3. Find the number of ways the Republicans can choose a subcommittee of 4 members. 4. Find the number of ways the Democrats can choose a subcommittee of 3 members. 5. Find the number of ways the Independents can choose a subcommittee of 3 members. 6. Find the number of ways the Independents can choose a Chairperson, Vice-chairperson, and Secretary, in this order. 7. Find the number of ways the Republicans can choose a Chairperson, Vice-chairperson, and Secretary, in this order. 8. Find the number of ways the Democrats can choose a Chairperson, Vice-chairperson, and Secretary, in this order.

Answer 1

1) 1320 2)278256 3)330 4) 455 5)56 6)336 7) 990 8) 2730

Explanation:

Solution:

1) Democrats = 15, Republicans = 11, Independents= 8

Total = 15+ 11+ 8 =34

Number of 3 persons committee = 15nCr 1 * 11nCr1 * 8nCr1

nCr =combination

Number of 3 persons committee that can be made by choosing any republican or democrat or independent = 15 * 11* 8 =1320

2)Total number of possible 5 person subcommittes from any 34 people = 34nCr5 = 34!/(29! * 5!)

=278256

3) Number of ways republicans choose subcomittee of 4 members = 11nCr4

= 11!/(7! * 4!)=330 ways

4) Number of ways democrats choose subcomittee of 3 members = 15nCr3

= 15!/ (12! *3!)=455 ways

5) Number of waysindependents choose subcomittee of 3 members = 8nCr3

= 8!/(5! * 3!)= 56 ways

6) Number of ways independents choose chairperson, vice chairperson and secretary:

As one picks each post the number of person decreases

= (Chairperson)* (vice chairperson) * (secretary) = 8nCr1 * 7nCr1 * 6nCr1

=336 ways

7) Number of ways republicans can choose chairperson, vice chairperson and secretary

As one picks each post the number of person decreases

= (Chairperson)* (vice chairperson) * (secretary) = 11nCr1 * 10nCr1 * 9nCr1

= 990ways

8) Number of ways Democrats can choose chairperson, vice chairperson and secretary

As one picks each post the number of person decreases

= (Chairperson)* (vice chairperson) * (secretary) = 15nCr1 * 14nCr1 * 13nCr1

= 2730 ways