# Counting

• Mar 26th 2007, 08:20 PM
AtoyotA
Counting
There are 3 women (A, B, C) and 3 men (D, E, F). How many ways can the 6 of them line up such that:

1.) No restrictions apply?

2.) Men and women have to alternate?

3.) First and last person in the line are women?

4.) A and D will not stand next to each other?

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I'm not sure how to go about these counting questions.

For #1, I believe it'd just be 6!, right?
• Mar 26th 2007, 09:43 PM
earboth
Quote:

Originally Posted by AtoyotA
There are 3 women (A, B, C) and 3 men (D, E, F). How many ways can the 6 of them line up such that:

1.) No restrictions apply?
2.) Men and women have to alternate?
3.) First and last person in the line are women?
4.) A and D will not stand next to each other?
...

Hello,

to 1.): You have 6 places to fill out of 6 persons. For the first place you have 6 possibilities, for the second place you have 5 possibilities, ... in the end you have totally 6*5*...*1 = 6! = 720 possibilities.

to 2.): First you have to decide which gender starts the line: 2 possibilities. For the first place you have 3 possibilities, for the second place you have 3 possibilities, for the third place you have 2 possibilities, ... in the end you have totally 2 * 3! * 3! = 72 possibilities

to 3.): You have 4 places to fill out of 4 persons. According to the considerations of #1.) you have totally 4! = 24 possibilities.

to 4.): There are 2 different cases:
a) A takes the first place then you have 4 possibilities to place D. The same situation occurs if A takes the last place. In total: 2 * 4 = 8 possibilities.

b) A take the second to 5th place then you have 3 possibilities to place D. In total: 4 * 3 = 12 possibilities

There are 20 possibilities so that A and D don't stand together.

EB

• Mar 27th 2007, 12:08 AM
AtoyotA
Thanks for the great explanation earboth.

Couple more questions, if I may. If we had to use the same 6 people, how would I count how many committees of 3-people are possible with:

1.) No restrictions?

2.) At least 1 woman?

3.) With B and C not on the same committee while A and D will only serve together?

Many thanks!!
• Mar 27th 2007, 05:33 AM
earboth
Quote:

Originally Posted by AtoyotA
Thanks for the great explanation earboth.

You're welcome!

Quote:

Couple more questions, if I may. ...
Hello,

But do yourself and do us a favour: If you have new questions please start a new thread. Otherwise nobody here will notice that you need some additional help.

EB