Final answer:
To determine the number of possible committees with 3 Republicans and 3 Democrats from the US Senate, calculate the combinations for each party and multiply them, resulting in 382,248,000 possible committees.
Step-by-step explanation:
To determine the number of committees that can be formed with 3 Republicans and 3 Democrats from the US Senate of the 104th Congress, which consists of 52 Republicans and 48 Democrats, we can use combinatorics. The number of ways to choose 3 Republicans out of 52 is calculated using combinations, denoted as C(52, 3). The number of ways to choose 3 Democrats out of 48 is denoted as C(48, 3).
To find the total number of different committees, we multiply the number of ways to choose the Republicans by the number of ways to choose the Democrats:
C(52, 3) × C(48, 3)
We use the combination formula C(n, k) = \( \frac{n!}{k! (n-k)!} \) where \( n! \) is the factorial of n, which is the product of all positive integers up to n.
Calculating the factorials and applying the formula, we get:
C(52, 3) = \( \frac{52!}{3! (52-3)!} =\frac{52 × 51 × 50}{3 × 2 × 1} = 22,100
C(48, 3) = \( \frac{48!}{3! (48-3)!} =\frac{48 × 47 × 46}{3 × 2 × 1} = 17,296
So the total number of committees is 22,100 × 17,296 = 382,248,000 committees.