Out of 6 teachers and 4 boys, a committee of 8 is to be formed . In ho...
**Problem:**
Out of 6 teachers and 4 boys, a committee of 8 is to be formed. In how many ways can this be done when there should not be less than four teachers in the committee?
**Solution:**
To solve this problem, we need to use the concept of combinations.
**Step 1:** Determine the number of ways to select 4 or more teachers from the 6 teachers.
- We can select 4 teachers from 6 in C(6,4) ways.
- We can select 5 teachers from 6 in C(6,5) ways.
- We can select 6 teachers from 6 in C(6,6) ways.
**Step 2:** Determine the number of ways to select the remaining members of the committee.
- We need to select 4 members from the remaining 8 (4 boys and the selected teachers) in C(8,4) ways.
**Step 3:** Multiply the results from Step 1 and Step 2 to get the total number of ways to form the committee.
- For each case in Step 1, we have C(8,4) ways to select the remaining members.
- Therefore, the total number of ways to form the committee is:
C(6,4) * C(8,4) + C(6,5) * C(8,3) + C(6,6) * C(8,2)
**Step 4:** Simplify the result.
- C(6,4) = 15
- C(6,5) = 6
- C(6,6) = 1
- C(8,4) = 70
- C(8,3) = 56
- C(8,2) = 28
- Therefore, the total number of ways to form the committee is:
15 * 70 + 6 * 56 + 1 * 28 = 1186
**Answer:**
There are 1186 ways to form the committee with at least 4 teachers.
Out of 6 teachers and 4 boys, a committee of 8 is to be formed . In ho...
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