A committee of 6 U.S. senators is to be formed with 3 Democrats and 3 Republicans. In how many ways can this be done if there are 32 Democratic senators and 68 Republican senators?

We need to form a committee of 6 U.S. senators, consisting of 3 Democrats and 3 Republicans. Given that there are 32 Democratic senators and 68 Republican senators, we will calculate the number of ways to choose 3 Democrats and 3 Republicans from these groups.

### Step 1: Choose 3 Democrats from 32 Democratic Senators
The number of ways to choose 3 Democrats from 32 is given by the combination formula:

\[
\binom{32}{3} = \frac{32 \times 31 \times 30}{3 \times 2 \times 1} = 4960
\]

### Step 2: Choose 3 Republicans from 68 Republican Senators
Similarly, the number of ways to choose 3 Republicans from 68 is:

\[
\binom{68}{3} = \frac{68 \times 67 \times 66}{3 \times 2 \times 1} = 50116
\]

### Step 3: Total Number of Ways to Form the Committee
To find the total number of ways to form the committee, we multiply the number of ways to choose the Democrats and the Republicans:

\[
\text{Total ways} = \binom{32}{3} \times \binom{68}{3} = 4960 \times 50116 = 248576960
\]

Thus, the total number of ways to form the committee is \( 248,576,960 \).

Would you like more details on combinations or any part of the solution?

Here are five related questions you can explore:
1. How would the number change if we were to select 4 Republicans and 2 Democrats?
2. What if the committee needed to include at least one independent senator?
3. How many ways can a committee of 4 Democrats and 4 Republicans be formed?
4. What if we chose all senators randomly from the full 100-member senate?
5. How do combinations differ from permutations in such problems?

**Tip:** When solving combination problems, remember that order doesn't matter, but in permutation problems, order does.

Discover how to form a committee of 6 U.S. senators, including 3 Democrats and 3 Republicans. Learn about combinations and how to calculate the number of ways to choose senators from groups of 32 Democrats and 68 Republicans. Perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation