Question

A country has two political parties, the D and the R. Suppose that the national senate consists of 100 members, 44 of which are Ds and 56 of which are Rs.

How many ways are there to select a committee of 10 senate members with the same number of Ds and Rs?

Answer 1

There are 4,149,054,848,128 ways to form a committee of 10 senate members with an equal number of Ds and Rs, from a senate consisting of 44 Ds and 56 Rs.

Step-by-step explanation:

To determine the number of ways to select a committee of 10 senate members with an equal number of Democrats (D) and Republicans (R) from a senate consisting of 44 Ds and 56 Rs, we use combinations. Since the committee must have equal numbers of Ds and Rs, we are selecting 5 members from each group. The number of ways to select 5 Ds from the 44 is represented by the combination formula C(n, k) = \( \frac{n!}{k!(n-k)!} \), where in this case n is 44 and k is 5.

Similarly, to select 5 Rs from the 56, we also use the combination formula with n being 56 and k being 5. The total number of ways to select the committee is the product of the two separate selections, which is C(44, 5) × C(56, 5).

Computing these:

• The number of ways to select 5 Ds is C(44, 5) = 1,086,008.
• The number of ways to select 5 Rs is C(56, 5) = 3,819,416.

Therefore, the total number of ways to form the committee is 1,086,008 × 3,819,416, which equals 4,149,054,848,128 possible combinations.