different orders can a photographer pose a row of six people

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In how many different orders can a photographer pose a row of six people withot having the tallest person behind the shortest?

I did this before: 5!*2=240
6!=720

720-240=480

The only problem is that I don't get why i did the *2 part? I can't find anything in my notes explaining that, so if anyone knows, please tell me

eii exam tomorrow lol
 
??? It's a row. Why is anyone behind anyone else?
 
Re: different orders can a photographer pose a row of six pe

Hello, anna!

Check the wording of the problem . . . it doesn't make sense.

In how many different orders can a photographer pose a row of six people
without having the tallest person behind the shortest?
Click to expand...

According to the answer, it should be "... the tallest person next to the shortest."

Duct-tape the tallest and shortest together.
. . Then we have five "people" to arrange: \(\displaystyle \:\{\fbox{AB},\,C,\,D,\,E,\,F\}\)
They can be arranged in \(\displaystyle \,5!\) ways.

But the taped pair can ordered, too: \(\displaystyle \:\{\fbox{BA},\,C,\,D,\,E,\,F\}\)
. . This provides \(\displaystyle \,5!\) more ways.

Hence, there are: \(\displaystyle \:2\,\times\,5!\:=\:240\) ways where the tallest and shortest are adjacent.

Got it?

 
Shouldn’t it be ‘beside’ and not behind?
Without having the shortest beside the tallest.
In other words, we want to separate the shortest and the tallest.
There are four other people to do the separating, creating five places to put the two.
Choose two places from five; arrange the four others; arrange the two:
\(\displaystyle \L { 5 \choose 2} \left( {4!} \right)\left( {2!} \right)\).
 
I think that "behind" means "together"(side by side).

Now assume, you tied the shortest and the tallest person - together (may illegal in some countries - but do it anyway - just thought experiments).

Arrangement 1:

Tallest to the right and shortest to the left.

You can arrange this in 5! ways (with other "midling" four people).

Arrangement 2:

Then you can put Tallest to the left and shortest to the right.

You can arrange this in 5! ways.

In total, you can arrange 6 people where tallest and the shortest persons are side by side (arrangement 1 and 2) in (5! + 5! = )5! * 2 ways
 
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