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#1 2006-12-23 03:52:13

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Tricky arrangement problem

In how many ways can the letters of the word PENCILS be arranged in a row so that the L precedes the N? (Answer = 2520)

I am... really confused with this one.

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#2 2006-12-23 03:55:38

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Tricky arrangement problem

There only exist two situation , L precdeds the N , or N precedes the L  ,that means there are only half of the arrangement satisfy the need ,   the total arrangements 7! ,  then the satifying arrangement  is  7!/2 =2520


Numbers are the essence of the Universe

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#3 2006-12-23 04:01:25

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

Re: Tricky arrangement problem

Oh... I see. Thanks Stanley, I hadn't noticed that.

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#4 2006-12-23 04:08:52

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Tricky arrangement problem

you're welcome. I just happen to have an inspiration,hehe

Last edited by Stanley_Marsh (2006-12-23 04:10:12)


Numbers are the essence of the Universe

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#5 2006-12-23 05:11:12

Devantè
Real Member
Registered: 2006-07-14
Posts: 6,400

Re: Tricky arrangement problem

You'd be surprised how much factorials pop up.

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#6 2006-12-24 06:41:36

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

Re: Tricky arrangement problem

That was a tricky one.
It gave me inspiration,too.
Try with the word PENCILLS.
I can rephrase the Stanley's solution:
Let S be the set of all words.
We have 2 sets of words:
A)..xxNx..xLxx..
B)..xxLx..xNxx..
Then A U B = S and A . B = {} (here "." stands for 'or')
And, there exists a bijection θ: A->B, so |A|=|B|.
We can use simular observations to prove the following theorem:
Let

is a word with length m, in which the characters
are met once. Let
be some permutation of {1,2,...,n}. Then the number of words, rearranged from 
, in which
precedes
and
precedes
and ... and
precedes
, is:
,
where num(W) means the total number of words (which is not everytime equal m!).

Last edited by krassi_holmz (2006-12-24 06:44:14)


IPBLE:  Increasing Performance By Lowering Expectations.

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