in a club there are 6 men and 9 women. a committee of 1 man and 2 women is to be formed. how many different possibilities would there be in choosing the committee?
To solve this problem, we can use the concept of combinations. A combination represents the number of possibilities we have when selecting a specified number of objects from a larger set without regard to their order.
In this case, we need to select 1 man out of 6 and 2 women out of 9.
The number of possibilities for selecting 1 man out of 6 can be calculated using the formula for combinations, denoted as C(n, r), which represents the number of ways to choose r objects from a set of n objects. The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where "!" denotes the factorial of a number, which is the product of all positive integers up to that number.
Using this formula, we calculate the number of possibilities for selecting 1 man out of 6:
C(6, 1) = 6! / (1!(6-1)!) = 6! / (1! * 5!) = (6 * 5 * 4 * 3 * 2 * 1) / (1 * 5 * 4 * 3 * 2 * 1) = 6
So, there are 6 different possibilities for selecting 1 man from the group.
Similarly, the number of possibilities for selecting 2 women out of 9 can be calculated as follows:
C(9, 2) = 9! / (2!(9-2)!) = 9! / (2! * 7!) = (9 * 8 * 7!)/(2 * 1 * 7!) = 36
Now, to find the total number of possibilities for selecting the committee of 1 man and 2 women, we multiply the number of possibilities for each category:
Total possibilities = number of possibilities for selecting 1 man * number of possibilities for selecting 2 women
Total possibilities = 6 * 36 = 216
Therefore, there are 216 different possibilities for choosing the committee.