5lb book, Ch. 23, pg. 806, Q #47

[Jmoustafa345]

Hello! So looking at the answer explanation for #47, I understand first finding 5! and then using 4! to find the number of possibilities of having Beth and Dan lined up next to each other so that we can subtract that from 5!. What I don't understand is why we are doubling 4!. In the explanation, it says that we double 4! because there are actually two options: Beth first, then Dan, and Dan first, then Beth. But doesn't 4! include both options already, Beth/Dan and Dan/Beth? My understanding is that all possible options are included in x!, and that is why, if order doesn't matter, you divide x! by the factorial of the number of slots. When I worked this problem, I actually got D) 96.

[tommywallach]

Nope! I don't like the way they explain it at all. (I don't like that 4).

Instead, think of it this way, with 5 slots like so:

_ _ _ _ _

Now, let's imagine Beth and Dan in the first two slots:

B D _ _ _

Now, how many ways are there to arrange the other three slots?

B D 3 2 1 --> 3 * 2 * 1 = 6 ways

Now, how many places could Beth and Dan stand together?

B D _ _ _
_ B D _ _
_ _ B D _
_ _ _ B D

So that's 4 places. Let's multiply what we have so far:

6 * 4 = 24

But we have to remember that in every situation so far, it's been B before D. We need to multiply by 2 for the switcheroo, D before B in every listed order.

24 * 2 = 48

That makes a lot more sense to me, anyway!

-t

[Jmoustafa345]

Ah, major lightbulb went off. Thank you!

[tommywallach]

Glad to help!

-t