Questions: Question 16 1.3 39 Homework: Lesson 18 Homework The Senate in a certain state is comprised of 58 Republicans, 39 Democrats, and 3 Independents. How many committees can be formed that contain 3 Republicans, 2 Democrats, and 1 Independent? 3066 committees can be formed. Points: 0 of 1 Save

Solution Steps

To solve this problem, we need to calculate the number of ways to choose 3 Republicans from 58, 2 Democrats from 39, and 1 Independent from 3. This is a combinatorial problem where we use the combination formula, often denoted as C(n, k) or "n choose k", which calculates the number of ways to choose k items from a set of n items without regard to the order of selection. The total number of committees is the product of these individual combinations.

To find the number of ways to choose 3 Republicans from 58, we use the combination formula:

\[ \binom{58}{3} = \frac{58 \times 57 \times 56}{3 \times 2 \times 1} = 30,856 \]

To find the number of ways to choose 2 Democrats from 39, we use the combination formula:

\[ \binom{39}{2} = \frac{39 \times 38}{2 \times 1} = 741 \]

To find the number of ways to choose 1 Independent from 3, we use the combination formula:

\[ \binom{3}{1} = \frac{3}{1} = 3 \]

The total number of committees is the product of the combinations calculated in the previous steps:

\[ \text{Total Committees} = \binom{58}{3} \times \binom{39}{2} \times \binom{3}{1} = 30,856 \times 741 \times 3 = 68,592,888 \]

Final Answer