In how many ways can a subcommittee of six be chosen from a Senate committee of four Democrats and six Republicans if
(a) All members are eligible?
(b) The subcommittee must consist of two Republicans and four Democrats?
To solve this problem, we can use combinations, which is a concept in combinatorics that deals with selecting a certain number of items from a larger set without regard to their order.
(a) To find the number of ways to choose a subcommittee without any restrictions, we can simply choose any six individuals from the total of ten committee members, regardless of their party affiliation.
The total number of ways to do this is denoted by the combination formula:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of items to choose from, and k is the number of items to be chosen.
In this case, n = 10 (total committee members) and k = 6 (number of subcommittee members). Therefore, we can calculate the answer:
C(10, 6) = 10! / (6!(10-6)!) = 10! / (6!4!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
Therefore, there are 210 ways to choose a subcommittee of six members from the Senate committee of four Democrats and six Republicans if all members are eligible.
(b) Now let's consider the constraint that the subcommittee must consist of two Republicans and four Democrats.
To solve this, we can break it down into two steps:
Step 1: Choose two Republicans from the six available:
C(6, 2) = 6! / (2!(6-2)!) = (6 * 5) / (2 * 1) = 15
Step 2: Choose four Democrats from the four available:
C(4, 4) = 4! / (4!(4-4)!) = 1
To find the total number of ways, we multiply the results from both steps together:
15 * 1 = 15
Therefore, there are 15 ways to choose a subcommittee of six members from the Senate committee of four Democrats and six Republicans if the subcommittee must consist of two Republicans and four Democrats.