# Selection using permutation and combination

From 4 men and 4 ladies a committee of 5 is to be formed. The committee consists of a president, vice president and three secretaries.

What will be the number of ways of selecting the committee with atleast 3 women such that atleast one woman holds the post of either a president or vice president.

I have been staring at this question for quite long; not sure where to start.

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There are several ways to approach this.

One is to divide it into cases, depending on how many women there are.

1. If there are 4 women in the committee, how many ways can you select the man? (The women are "forced", since there are only four women). How many ways can you assign roles? Well, select a president, then a vice president (the rest are secretaries). Will all possible choices work, or are there some "forbidden" choices?

2. If there are exactly three women in the committee. You have 4 ways of selecting the women to be in the comittee; and $\binom{4}{2}=6$ ways of selecting the men to be in the committee. As to positions, there is only one combination we don't want, which is when the men occupy both the presidency and the vice-presidency. How many total ways are there to assign roles? How many of those are bad? Subtract the bad from the total, then calculate.

Finally, add the number of ways of forming the committee with three women, and of forming the committee with four women.

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Break it down. First, the committee can have $3$ or $4$ women. How many ways are there to choose the committee so that it has $3$ women? How many ways so that it has $4$ women?

Now suppose that I’ve chosen my committee. How many ways are there to assign officer positions to the committee members? There are $5$ ways to choose the president; once that’s been done, there are $4$ ways to choose the vice president; and once that’s been done, everyone else is a secretary. Thus, there are $5\cdot4=20$ ways to assign officer positions.

Now suppose that I’ve chosen a committee with $3$ women and $2$ men. In how many ways can I assign officer positions so that both the president and the vice president are men? These are the unwanted ways of doing it; subtract this number from $20$ to get the number of committees in which a woman holds at least one of the top two positions.

Finally, suppose that I’ve chosen a committee with $4$ women and one man. How many of the officer assignments will give at least one of the top two positions to a woman?

Now put the pieces together.

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Break it down by cases: Count the number of ways with 4 women in the committee first. This is easy, since the extra condition is automatically satisfied. Then count the number of ways with 3 women, disregarding the extra condition. Count and subtract the number of committees in which all the women are secretaries.

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Hint 1: How many ways to choose a committee of $5$ from these $8$ people such that at least $3$ women are on the committee?

Hint 2: Of those, how many such will have a male president and a male vice president? (Recall that there must be at least three women!)

If we have a committee of $4$ women, there are only two questions to answer: Which positions can we put the man in? Which man has that position? Well, either the man committee member is pres., vice pres., or secretary, so there are $3$ positions the man can end up in. Also, there are $4$ men who could take that position. Thus, there are $$3\cdot 4=12$$ such committees with only one man. Clearly, we can't have an all-woman committee (since there are only $4$ of them), and we are required to have at least $3$, so now we have to determine how many $3$-woman committees there are. It's still easier to work with the men, since there will be fewer of them, and again, we've got two questions to answer: Which positions can the two men can be placed distinctly in? How many ways can we pick the two men? There are $4$ men, of whom $2$ are chosen for the committee, so that there are $$\binom 4 2=6$$ ways to pick the men for the committee. Suppose we've picked them, and need only assign positions--let's call them Al and Bob, for distinction. We could have: both of them as secretaries; Al as pres. and Bob as a sec.; Al as vice pres. and Bob as a sec.; Al as pres. and Bob as vice pres.; Bob as pres. and Al as a sec.; Bob as vice pres. and Al as a sec.; Bob as pres. and Al as vice pres.--in total, that's $7$ different ways that the ranks can be divided among the two chosen men. Thus, there are $$6\cdot 7=42$$ such committees with only two men, and so there are $54$ committees with at least three women. However, some of those had females as secretaries only, which we don't want. Well, how many of those were there? It only happened with the $2$-man committee, and only when one was pres. and the other was vice pres. There were $6$ ways to pick the two men, and $2$ ways to rank them as pres. and vice pres., so that's $$6\cdot 2=12$$ committee configurations to discard, leaving $$54-12=42$$ committees of the type we wanted.