To solve the problem of determining the number of different 9-member committees possible with exactly 4 Biology teachers, we need to follow these steps:

Step-by-Step Solution:

1. Total Teachers Available:

   • Mathematics teachers = 8
   • Chemistry teachers = 10
   • Biology teachers = 13

2. Committee Requirements:

   • Total committee members = 9
   • Biology teachers required = 4
   • Remaining members to be selected from Mathematics and Chemistry = $9 - 4 = 5$

3. Calculate Combinations:

   • Choose 4 Biology Teachers:

     • The number of ways to choose 4 Biology teachers out of 13 is given by the combination formula:

     $\binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715$

   • Choose 5 Teachers from Mathematics and Chemistry:

     • Total teachers available from Mathematics and Chemistry = $8 + 10 = 18$
     • The number of ways to choose 5 teachers from these 18 teachers is given by:

     $\binom{18}{5} = \frac{18!}{5!(18-5)!} = \frac{18 \times 17 \times 16 \times 15 \times 14}{5 \times 4 \times 3 \times 2 \times 1} = 8568$

4. Calculate Total Number of Committees:

   • Multiply the number of ways to choose the Biology teachers by the number of ways to choose the remaining teachers:

     $\text{Total Committees} = \binom{13}{4} \times \binom{18}{5} = 715 \times 8568 = 612,612$

Therefore, the number of different 9-member committees possible that include exactly 4 Biology teachers is 612,612.

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

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Related Questions:

1. How many different 9-member committees are possible if no specific requirements on the number of Biology teachers are given?
2. If 5 committee members must be Biology teachers instead of 4, how many such committees can be formed?
3. What if the committee needs to include at least one teacher from each subject?
4. How many ways can you select a 7-member committee with equal representation from all three subjects?
5. How would the answer change if there were 15 Biology teachers instead of 13?
6. How many different committees can be formed if the committee size is increased to 11 members with the same Biology teacher requirement?
7. What if there must be at least 6 Mathematics teachers on the committee?
8. How many ways can we select a subcommittee of 4 teachers from a committee already formed under these conditions?

Tip:

When solving combination problems, carefully identify how many choices are needed from each group and use the combination formula $\binom{n}{r}$ to calculate the number of possible selections.