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.

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.