Hello, all! I need some help with a certain question:

The letters A, B, C, D, E and F are to be arranged in a straight line.

How many ways are there of arranging the six letters in a straight line if A is before B?

Question

Hello, all! I need some help with a certain question:

The letters A, B, C, D, E and F are to be arranged in a straight line. 

How many ways are there of arranging the six letters in a straight line if A is before B?

inkyvoyd · Answer

there's a few ways to do this, but let's think about this for a minute. Either A is before B in the arrangement or A is after B in the arrangement, correct?

anonymous · Answer

Yes, then?

inkyvoyd · Answer

well, are there more ways to arrange A before B, or are there more ways to arrange B before A? Or do they seem to be equal?

anonymous · Answer

I believe that there's more ways that A can be before B.

anonymous · Answer

As compared to B being before A, since the only way would be to have A as the last one.

inkyvoyd · Answer

hmm. Imagine that you have a list of all the ways that A can be before B. Well, it turns out you can switch B with A.... and then you have a list of all the ways that B can be before A

anonymous · Answer

Oh, then it'd be equal? 

(And lol, I just realized my previous response was stupid, there's also the XXBAXX option on top of many more.)

inkyvoyd · Answer

yes... so if in all the arrangements either A must be before B or B must be before A, and the ways to arrange A before B and B before A are equal, doesn't that just mean that the number of ways of arranging A before B is merely half the total?

anonymous · Answer

Aha. Got it. Thank you very much!

inkyvoyd · Answer

no problem :)