# Answer to sister in PJ’s questions

Without the loss of generality, we can start out with Chloe having first dibs on the outfit. It doesn’t matter which sister we pick. Let’s also have Chloe choose a top first, and then a mismatched bottom.  She has 3 choices for a top: star, elephant, or bow. If she wears a star top, she can wear an elephant or bow bottom. Similarly, if she wears an elephant top, she can wear a star or a bow bottom. Finally, if she wears a bow bottom, she can wear a star or elephant top. So Chloe has a choice of six (6) different outfits. You can see them all in the first column of the diagram.

Follow lines from left to right go get an assignment of outfits.

Once Chloe chooses an outfit, how many outfits are left for her sisters to wear? It turns out there are only two. We can see this in the following diagram:

In the example above, Chloe is wearing a star shirt and bow pants. Is it possible for either Sophie or Grace to wear a bow shirt and star pants? At first it may seem like the answer might be “yes”. However, if this happens, what is outfit is left over? An elephant top and an elephant bottom. But the sisters may only wear mismatched pajamas.

Moral of the story: once Chloe wears a star shirt and bow pants, no other sister may wear a bow shirt and star pants.

Now that we have eliminated this combination, we see that the only allowable outfits left for Sophie and Grace are:

• Bow shirt and elephant pants (in red oval)
• Elephant shirt and star pants (in blue oval)

We can either have Sophie wear outfit (1) and Grace wear outfit (2), or have Sophie wear outfit (2) and Grace wear outfit (1). So for every outfit Chloe chooses, there are 2 possible arrangements left for her sisters. Since Chloe has a choice of 6 outfits, all together there are 6 X 2 =12 possibilites.