In how many ways can a comittee of 3 women and 4 men be chosen from 8 women and 7 men if two particular women refuse to serve on the committe together? Answer is 1750 but I have no clue how to get to it
Let's choose 3 women and 4 men from the 8 women and 7 men without any restrictions.
= C(8,3) x C(7,4) = ....
Now choose with the two particular women in place and th 4 men from the 7
That means we have to choose 1 more women from the remaining 6, the men situation does not change
= C(6,1) x C(7,4)
subtract this from the total found first, that should give you 1750
(worked for me)
men ... 7C4 possibilities ... 35
women ... 8C3 possibilities ... 56
... but 6 of the groups can't be used ... the incompatible pair with any of the other 6 women
35 * (56 - 6) = ?
To find the number of ways to choose a committee of 3 women and 4 men with the given conditions, we can break down the problem into smaller steps.
Step 1: Choose 3 women from the 8 available.
Since two particular women refuse to serve on the committee together, we need to consider two cases:
- Case 1: The two particular women both serve on the committee.
- Case 2: The two particular women do not both serve on the committee.
In Case 1, we have 2 women already selected. So we need to choose 1 more woman from the remaining 6 women (excluding the 2 particular women).
In Case 2, we need to choose all 3 women from the remaining 6 women (excluding the 2 particular women).
Step 2: Choose 4 men from the 7 available.
Finally, to calculate the total number of ways, we need to add the number of ways from Case 1 and Case 2.
Let's calculate the number of ways for each case separately:
Case 1: The two particular women both serve on the committee.
Number of ways = (Number of ways to choose 1 woman from the remaining 6 women) * (Number of ways to choose 4 men from the 7 available)
= C(6, 1) * C(7, 4)
= 6 * 35
= 210
Case 2: The two particular women do not both serve on the committee.
Number of ways = (Number of ways to choose all 3 women from the remaining 6 women) * (Number of ways to choose 4 men from the 7 available)
= C(6, 3) * C(7, 4)
= 20 * 35
= 700
Total number of ways = (Number of ways in Case 1) + (Number of ways in Case 2)
= 210 + 700
= 910
Therefore, there are 910 different ways to choose a committee of 3 women and 4 men from the given conditions.