For The Permutation Lovers (1 Viewer)

[Ilovecarrots]

In how many ways can a boat crew of eight women be arranged if three of the women can only row on the bow side and two others can only row on the stroke side?

Answer is 1724. Can someone quickly explain why? Thanks

 

[fan96]

The answer is 1728 (it's from the Cambridge book, I looked it up myself).



There are three "free" women who can row anywhere on the boat.

Consider the bow and stroke sides separately.

For the bow side, three specific women MUST be there, so one spot is left for one of the free women.

For the stroke side, two specific women MUST be there, so two spots are left for two of the free women.

There are ways to choose two of the three free women to sit on the stroke side (the remaining one then must sit on the bow side).

Then there are ways to arrange the bow side and then another ways to arrange the stroke side.

 

[Roy G Biv]

As always with restrictions questions, we will deal with the restrictions first:

the 3 gals on the bow side:
4 places for the first chick, 3 places for the second, 2 places for the last.
i.e. 4 x 3 x 2

the 2 gals on the other side:
4 places for the first chick, 3 places for the second.
i.e. 4 x 3

Now that we're done with the restrictions, we note that 3 spots will be open. The remaining 3 chicks can be placed in any of the remaining 3 slots:
3 places where we can place the first chick, 2 places for the second, 1 for the last.
i.e. 3 x 2 x 1

so, multiplying all of that:
4 x 3 x 2 x 4 x 3 x 3 x 2 x 1 = 1728 as required.

 

[HeroWise]

Here is my crack

 

[Ilovecarrots]

Thank you to everyone who replied! Highly appreciated