I have started learning permutation and combination. I am looking at the question below. It has given the answers also but I didn't understand how? I have looked at many examples online it just made me more confused. Can anyone briefly explain the concept, please?
A club with $20$ women and $17$ men needs to choose three different members to be president, vice president, and treasurer.
- In how many ways is this possible?
- In how many ways is this possible if women will be chosen as president and vice president and a man as treasurer?
Answer: (a) 37 . 36 . 35. (b) 20 . 19 . 17.
Best Answer
When you ask "in how many ways can we choose three people to be [three different roles]" it makes a difference which person is in which roll. Thus naming the men Adam, Ben, Charlie... the choice "Adam president, Ben vice president, Charlie treasurer" is distinct from the choice "Adam president, Charlie vice president, Ben treasurer."
With this in mind, we can choose any one of the 37 people (women or men) to be president. Now if I fix who I choose as president (say Terry) how many ways can I choose the vice president? It would be 37, but one of them -- Terry -- is not a "different member" from the president, so there are only 36 choices. So there are $37\cdot 36$ ways to choose the pair (president, vice president)": 36 with Adam as president,+ 36 with Ben as president $\cdots$ + 36 with Terry as president.
Now choose a treasurer; by similar reasoning there are only $35$ ways to do this, and this leads to a total number of choices for the three positions of $37\cdot 36 \cdot 35$.
For part (b) the reasoning is the same, only now there were only $20$ choices for president, and having eliminated the chosen president, only 19 choices for vice president. And then the treasurer has to be a man, which gives only 17 choices. Thus $20\cdot 19 \cdot 17$.
Pretty soon, you will be asked questions like "in how many ways can the whole slate of three different people be chosen, disregarding the specifics of which one of the three is president, which is vice president, and which is treasurer. You will find that you have to divide your answer to (a) by $6$ to answer that question, because there are six position-specific choices possible for each selection of three candidates.
And you will be asked questions like "how many ways can a slate be chosen that includes 2 women and 1 man?" Here you have to divide the answer to (b) by 2, because "Ann president, Terry vice president, some man treasurer" is the same slate as " "Terry president, Annvice president, same man treasurer."
The first type of question looks at "permutations", the second type at "combinations."