In how many ways can a committee of

fourfour

Democrats and

fourfour

Republicans be formed from a group of

tenten

Democrats and

twelvetwelve

​Republicans?

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Part 1

A committee of

fourfour

Democrats and

fourfour

Republicans can be formed from a group of

tenten

Democrats and

twelvetwelve

Republicans in

enter your response here

different ways.

To solve this problem, we need to determine the number of ways to select 4 Democrats from a group of 10 and 4 Republicans from a group of 12.

### Step-by-step Calculation

1. **Select 4 Democrats from 10:**
   The number of ways to choose 4 Democrats from 10 is given by the binomial coefficient \(\binom{10}{4}\).

   \[
   \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!}
   \]

   Calculating this, we get:

   \[
   \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
   \]

2. **Select 4 Republicans from 12:**
   The number of ways to choose 4 Republicans from 12 is given by the binomial coefficient \(\binom{12}{4}\).

   \[
   \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!}
   \]

   Calculating this, we get:

   \[
   \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495
   \]

3. **Total Number of Ways to Form the Committee:**
   The total number of ways to form the committee is the product of the two binomial coefficients.

   \[
   \text{Total ways} = \binom{10}{4} \times \binom{12}{4} = 210 \times 495 = 103950
   \]

### Conclusion

The number of different ways to form a committee of 4 Democrats and 4 Republicans from a group of 10 Democrats and 12 Republicans is **103,950**.

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Would you like more details on any of these steps, or do you have any additional questions?

### Related Questions:

1. How would the number of ways change if the committee consisted of 5 Democrats and 3 Republicans?
2. What if there were only 8 Democrats and 15 Republicans to choose from?
3. How does the calculation change if the committee requires at least 3 members from each party?
4. How many ways can a committee of 4 Democrats and 5 Republicans be formed from the same group?
5. If the committee had no restrictions on party affiliation, how many total committees of 8 members could be formed?

**Tip:** When dealing with combinations, remember that order does not matter. This is a key characteristic that distinguishes combinations from permutations.

Learn how to calculate the number of ways to form a committee consisting of 4 Democrats and 4 Republicans from a group of 10 Democrats and 12 Republicans. This problem involves combinatorial mathematics and uses binomial coefficients. Suitable for advanced high school students (Grades 10-12).

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