## Combinatorics.Exclude when specific thing is before another.

For the discussion of math. Duh.

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jacksmack
Posts: 83
Joined: Wed Mar 21, 2012 2:18 am UTC
Location: Italy

### Combinatorics.Exclude when specific thing is before another.

Hi,

I have some problems with this exercise about combinatorics. Please can you help me to resolve that? :

Five guys: John, Jack, Greg, Matt, Eric, will be speaking at a meeting. How many ways can they take their turn without Jack speaking before John?

I have done my own reasoning, as follows:
I have considered the case without restrictions: 5! = 120 ways.
And then I considered John and Jack as unique group I can arrange in 4! = 24 ways. I haven't considered the arrangement 2! for that group because I consider only the case where Jack is before John and not vice versa.
But when I do the calculation: 120 - 24 = 96

that it is different from the answer given in the textbook. Where is the error? Maybe Do I have to interpretate the exercise in a different way?

many thanks!

elliptic
Posts: 34
Joined: Tue Aug 14, 2012 2:21 pm UTC
Location: UK

### Re: Combinatorics.Exclude when specific thing is before anot

Total number of speaker orderings N(total) = 5! = 120.

By symmetry, N(Jack before John) = N(John before Jack).

Therefore, N(John before Jack) = N(total) / 2 = 60.

Sizik
Posts: 1260
Joined: Wed Aug 27, 2008 3:48 am UTC

### Re: Combinatorics.Exclude when specific thing is before anot

jacksmack wrote:I have done my own reasoning, as follows:
I have considered the case without restrictions: 5! = 120 ways.
And then I considered John and Jack as unique group I can arrange in 4! = 24 ways. I haven't considered the arrangement 2! for that group because I consider only the case where Jack is before John and not vice versa.
But when I do the calculation: 120 - 24 = 96

By considering Jack and John as one "person" in the lineup, you're only counting the cases where they're sitting right next to each other. Thus, 24 is the number of arrangements where Jack is immediately before John, and doesn't include when Jack is two, three, or four slots ahead.
she/they
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