A student committee of size 13 is to be selected from 14 Arts and Science students, and 19 Business students.  How many different 13-member committees are possible if 4 committee members have to be Business students?

To solve this problem, we need to calculate the number of ways to form a 13-member committee such that 4 of the members are Business students and the remaining 9 are from Arts and Science students.

### Step-by-Step Solution:

1. **Total Students:**
   - Arts and Science students = 14
   - Business students = 19

2. **Committee Requirements:**
   - Total committee members = 13
   - Business students required = 4
   - Arts and Science students required = \(13 - 4 = 9\)

3. **Calculate Combinations:**
   - The number of ways to choose 4 Business students out of 19 is given by the combination formula:

     \[
     \binom{19}{4} = \frac{19!}{4!(19-4)!} = \frac{19 \times 18 \times 17 \times 16}{4 \times 3 \times 2 \times 1} = 3876
     \]

   - The number of ways to choose 9 Arts and Science students out of 14 is given by:

     \[
     \binom{14}{9} = \frac{14!}{9!(14-9)!} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002
     \]

4. **Calculate Total Number of Committees:**
   - Multiply the number of ways to choose Business students by the number of ways to choose Arts and Science students:

     \[
     \text{Total Committees} = \binom{19}{4} \times \binom{14}{9} = 3876 \times 2002 = 7751952
     \]

Therefore, the number of different 13-member committees possible, with exactly 4 Business students, is **7,751,952**.

If you have any questions or need further explanations, feel free to ask!

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### Related Questions:

1. How many different 13-member committees are possible if no specific requirements on the number of Business students are given?
2. If 5 committee members must be Business students instead of 4, how many such committees can be formed?
3. What if the committee needs to include at least one student from both groups?
4. How many ways can you select a 10-member committee with equal representation from Arts and Science and Business students?
5. How would the answer change if there were 20 Arts and Science students instead of 14?
6. How many different committees can be formed if the committee size is increased to 15 members with the same Business student requirement?
7. What if there must be at least 6 Business students on the committee?
8. How many ways can we select a subcommittee of 5 students from a committee already formed under these conditions?

### Tip:
When dealing with combinations, remember the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). This formula is essential for calculating the number of ways to choose a subset of \(r\) elements from a set of \(n\) elements without regard to order.

Learn how to calculate the number of different 13-member committees possible when 4 members must be Business students. Step-by-step solution using combinatorics and binomial coefficients. Suitable for high school students.

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