A committee is to be formed consisting of 7 members from 8 parents, 5 teachers and the principal. How many ways can the committee formed if the committee consists of not more than 4 parents?
0 parents: 0 ways (not enough people)
1 parent: 8 * 5C5 * 1C1
2 parents: 8C2 * 5C4 * 1C1 + 8C2 * 5C5
and so on
add them all up.
To find the number of ways to form the committee with not more than 4 parents, we can use the principle of inclusion-exclusion.
There are 8 parents to choose from and we want to choose at most 4.
1. Selecting 0 parents:
We need to choose 7 members from the 5 teachers and the principal (6 options).
Therefore, there are 6 ways to form the committee with 0 parents.
2. Selecting 1 parent:
We need to select 6 more members from the remaining 7 parents, 5 teachers, and the principal (13 options).
Therefore, there are 13 * 7 = 91 ways to form the committee with 1 parent.
3. Selecting 2 parents:
We need to select 5 more members from the remaining 6 parents, 5 teachers, and the principal (12 options).
Therefore, there are 12 * 6 = 72 ways to form the committee with 2 parents.
4. Selecting 3 parents:
We need to select 4 more members from the remaining 5 parents, 5 teachers, and the principal (11 options).
Therefore, there are 11 * 5 = 55 ways to form the committee with 3 parents.
5. Selecting 4 parents:
We need to select 3 more members from the remaining 4 parents, 5 teachers, and the principal (10 options).
Therefore, there are 10 * 4 = 40 ways to form the committee with 4 parents.
Now, we can add up all the cases:
6 + 91 + 72 + 55 + 40 = 264 ways to form the committee with not more than 4 parents.
To solve this problem, we can break it down into cases:
Case 1: Selecting 0 parents
In this case, we need to select all 7 members of the committee from the 5 teachers and the principal. Since these are the only options available, there is only one way to form the committee in this case.
Case 2: Selecting 1 parent
In this case, we need to select 6 more members from the remaining 7 parents, 5 teachers, and the principal. To solve this, we can use the combination formula. The number of ways to select 6 members from a group of 7 parents, 5 teachers, and the principal is given by 7C6 * (5+1)C6 = 7 * 6 * 1 = 42.
Case 3: Selecting 2 parents
In this case, we need to select 5 more members from the remaining 6 parents, 5 teachers, and the principal. The number of ways to select 5 members from a group of 6 parents, 5 teachers, and the principal is given by 6C5 * (5+1)C5 = 6 * 6 = 36.
Case 4: Selecting 3 parents
In this case, we need to select 4 more members from the remaining 5 parents, 5 teachers, and the principal. The number of ways to select 4 members from a group of 5 parents, 5 teachers, and the principal is given by 5C4 * (5+1)C4 = 5 * 5 = 25.
Case 5: Selecting 4 parents
In this case, we need to select 3 more members from the remaining 4 parents, 5 teachers, and the principal. The number of ways to select 3 members from a group of 4 parents, 5 teachers, and the principal is given by 4C3 * (5+1)C3 = 4 * 15 = 60.
Now, we can calculate the total number of ways to form the committee by summing up the numbers from each case:
1 + 42 + 36 + 25 + 60 = 164
Therefore, there are 164 ways to form the committee if the committee consists of not more than 4 parents.