A group of 9 workers decides to send a delegation of 3 to their supervisor to dicusss their grievences.
c.) If there are 4 women and 5 men in the group, how many delegations would include at least 1 women?
I know I have to use combination C(9,1)x C(9,4) but I'm lost after that? Please help?
There are 9 of them so a delegation of 3 can go in 9C3 = 84 ways. There are 5 men in them so "all men" delgations will be 5C3= 10.
hence 84-10 = 74 delegations will contain at least one lady.
So what you have done and heard, both are wrong.
The number of possible delegations without considering any restriction is C(9,3) = 84.
The number of possible delegations which do not include any woman is C(5,3) = 10.
So the answer should be: 84-10 = 74.
I don't know how you get 129, and why is the answer 112. If you write your solution, I could check it to find the flaw (or maybe my solution is flawed).
To find the number of delegations that include at least 1 woman, we need to consider two cases: delegations with exactly 1 woman and delegations with 2 or more women.
Case 1: Delegations with exactly 1 woman
To calculate the number of delegations with exactly 1 woman, we need to choose 1 woman from the 4 available and 2 workers from the remaining 8 (which includes both men and women). This can be done using combinations.
Number of combinations with exactly 1 woman = C(4, 1) * C(8, 2) = 4 * 28 = 112
Case 2: Delegations with 2 or more women
To calculate the number of delegations with 2 or more women, we need to exclude the case where there are no women in the delegation. Therefore, we subtract the number of all-male delegations from the total number of possible delegations.
Total number of possible delegations = C(9, 3) = 84
Number of all-male delegations = C(5, 3) = 10
Number of delegations with 2 or more women = Total number of possible delegations - Number of all-male delegations = 84 - 10 = 74
To find the total number of delegations that include at least 1 woman, we add the number of delegations with exactly 1 woman and the number of delegations with 2 or more women.
Total number of delegations with at least 1 woman = Number of combinations with exactly 1 woman + Number of delegations with 2 or more women
= 112 + 74 = 186
Therefore, there are 186 delegations that include at least 1 woman.