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## #1 2012-10-23 15:30:00

slowlearner
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### Combinations

I would like some help with this puzzle
Six boys separately climb a tree with nine big branches to get a large apple at the top but they all fall out of the tree before reaching the apple.
The first climber slips and hits branch A, then B and finally C before landing on the ground.
The next boy falls and hits branch D, then E and finally F and lands on the ground.
The third boys also has problems and hits branches B, then H, then E.
The fourth boy tries with no luck as well as he hits branches C, then F, and lastly branch I on the way down.
The fifth boy goes up but loses his grib and hits branch A, then E, and then G.
Finally the last boy goes up and just about gets his hand on the apple when he falls and hits EVERY branch on the way down.
My question is -  How many different possible orders are there for him to have been so unlucky and have done this?

I came up with 24 but my dad said he thinks it is more than that and to ask my math teacher.
I asked my math teacher and she said she didn't know but would use her "phone a friend" and get back to me but that was 2 days ago.
Any help on here?  Thanx.

Last edited by slowlearner (2012-10-23 17:16:21)

## #2 2012-10-23 18:33:54

bob bundy
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### Re: Combinations

hi slowlearner,

I'm also getting 24. Revising this to 36.  Oh dear.  I think it's more than that.

Back to the drawing board.

OK Best answer so far is 46.  Here they are:

ABCDHEGFI
ABCHDEGFI

ABCDHEFGI
ABCHDEFGI

ABCDHEFIG
ABCHDEFIG

DABCHEGFI
ABDCHEGFI
ABHCDEGFI

DABCHEFGI
ABDCHEFGI
ABHCDEFGI

DABCHEFIG
ABDCHEFIG
ABHCDEFIG

DABHCEGFI
ABDHCEGFI
ABHDCEGFI

DABHCEFGI
ABDHCEFGI
ABHDCEFGI

DABHCEFIG
ABDHCEFIG
ABHDCEFIG

DABHECGFI
ABDHECGFI
ABHDECGFI

DABHECFGI
ABDHECFGI
ABHDECFGI

DABHECFIG
ABDHECFIG
ABHDECFIG

DABHEGCFI
ABDHEGCFI
ABHDEGCFI

I haven't checked this list yet so if you spot a repeat or a missing, plaese let me know.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #3 2012-10-23 23:07:21

bob bundy
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### Re: Combinations

So how did I get that list?

1: tells us that A is followed by B (maybe others in between)

Ignoring C and D for the moment

3: tells us that the order continues ABHE

2: tells us F follows E so ABHEF

4: allows us to add I getting ABHEFI

So that order is certain.

But where to put C D and G?

D must occur before E so there are 4 possibilities for that. (I'm not counting C yet)

G must occur after E so that gives 3 possibiliities.

These two branches (D and G) are separated by E so these choices are independent of each other.  Thus, 4 x 3 = 12 possibilities.

The difficuly is to place C.  It must comes after B and before F, but the D and G placements make the possibilities harder to sort out.

What I did was to make several copies of the 12 arrangements I had got with the other 6 branches and then tried to place C.

Firstly, can C be in position 3?  There are a few ways for this.  I deleted the impossible ones.

Then, on a fresh set of 12, I tried C in position 4.

I continued in this way with C in position 5, then 6 then 7.  8 and above is not possible.

And that's the list I came up with.

Hope it is right.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #4 2012-10-24 03:15:59

slowlearner
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### Re: Combinations

Thanks Bob for your help.  You did a great job.  My teacher just informed me she got the same figures from a university math professor who is a friend of hers.  We both get a gold star.

Last edited by slowlearner (2012-10-24 03:42:17)