In how many ways can a sub-committee consisting of a chairman, a vice chairman and a Secretary together with four other members be selected from a group of twelve councillors?
Let's pick the chair, vice-chair and secretary.
Number of ways = 12*11*10 = 1320
which leaves 9 to choose from for the remaining members
which would be C(9,4)
number of ways to choose your group = 1320*C(9,4) = ....
answer the question
To find the number of ways to select a sub-committee consisting of a chairman, a vice chairman, a secretary, and four other members from a group of twelve councillors, we can use the concept of combinations.
The process to solve this problem involves several steps:
Step 1: Selecting the chairman.
Since we need to select one chairman from twelve councillors, there are twelve choices for the chairman.
Step 2: Selecting the vice chairman.
After selecting the chairman, we need to select one vice chairman from the remaining eleven councillors. There will be eleven choices for the vice chairman.
Step 3: Selecting the secretary.
After selecting the chairman and the vice chairman, we need to select one secretary from the remaining ten councillors. There will be ten choices for the secretary.
Step 4: Selecting the four other members.
Now that we have selected the chairman, vice chairman, and the secretary, we need to select four additional members. Since we have already selected three councillors, there are only nine councillors left to choose from. We need to select four councillors from this remaining pool.
The number of ways to select the four remaining members can be calculated using combinations. The formula for combinations is:
nCr = n! / (r! * (n-r)!)
Where n is the total number of elements and r is the number of elements to be selected. In this case, n = 9 (remaining councillors) and r = 4 (number of members to be selected).
Using the formula, we can calculate: 9C4 = 9! / (4! * (9-4)!)
Step 5: Calculate the final answer.
To find the total number of ways, we need to multiply the number of choices at each step. Therefore, the total number of ways to select the sub-committee will be:
12 * 11 * 10 * 9! / (4! * 5!)
Simplifying this expression will give us the final answer.
Note: The exclamation mark (!) represents the factorial of a number, which means multiplying all positive integers up to that number. For example, 4! = 4 * 3 * 2 * 1.
Let's calculate the answer using this formula:
12 * 11 * 10 * (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1 * 5 * 4 * 3 * 2 * 1)
After simplifying the expression, the final answer will give us the total number of ways to select the sub-committee.