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# How many different five-letter combinations can be created

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VP
Joined: 10 Jun 2007
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How many different five-letter combinations can be created [#permalink]  20 Jun 2007, 16:00
6. How many different five-letter combinations can be created from the word TWIST?
A. 5
B. 24
C. 60
D. 120
E. 720
Senior Manager
Joined: 06 Jul 2004
Posts: 474
Location: united states
Followers: 1

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Re: Combination [#permalink]  20 Jun 2007, 16:20
bkk145 wrote:
6. How many different five-letter combinations can be created from the word TWIST?
A. 5
B. 24
C. 60
D. 120
E. 720

I got 42 which is not among the answer choices.

This is how I solved it:

TWIST

if both T's are together, the number of ways = 4! = 24

when both T's are spaced they either are

TXTXX
or
TXXTX
or
TXXXT

in all three such cases, the total number of words = 3*3! = 18

total number of 5 letter words = 24+18 = 42.
_________________

for every person who doesn't try because he is
afraid of loosing , there is another person who
keeps making mistakes and succeeds..

Director
Joined: 30 Nov 2006
Posts: 591
Location: Kuwait
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I remember the formula from the Project GMAT book:

Number of combinations using all the M letters in the group with L repetitions = M!/L!

for TWIST = 5!/2! = 5x4x3x2!/2! = 60

P.S.: am I violating copy rights by posting the equation from the PRoject Gmat book ?!! I'm a little worried honestly
VP
Joined: 10 Jun 2007
Posts: 1464
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Kudos [?]: 142 [0], given: 0

60 is correct...
Can someone explain how to arrive M! / L! ? I know it is a formula, but would love to know how to arrive it.
Manager
Joined: 21 May 2007
Posts: 120
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These are best explained using a diagram (tree diagrams), which I currently cant.
Let me give it a shot without one.

If all 5 characters were distinct, there were 5! possibilities. However, 'T' is repeated twice. How many ways the 2 T's can be arranged relative to one another? = 2! = 2
For each of these 2 arrangements, the other 3 characters are repeated. Hence you divide the 5! by 2!.

I believe that was lame, but without pictorial aids, its tough to articulate it.

best,
parsifal
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Posts: 474
Location: united states
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Re: Combination [#permalink]  20 Jun 2007, 17:25
shoonya wrote:
bkk145 wrote:
6. How many different five-letter combinations can be created from the word TWIST?
A. 5
B. 24
C. 60
D. 120
E. 720

I got 42 which is not among the answer choices.

This is how I solved it:

TWIST

if both T's are together, the number of ways = 4! = 24

when both T's are spaced they either are

TXTXX
or
TXXTX
or
TXXXT

in all three such cases, the total number of words = 3*3! = 18

total number of 5 letter words = 24+18 = 42.

Guys, I think I got it. I missed a few calculations in my last post.

when both T's together, the total number of ways = 4! = 24

when they are apart, they are

apart by one space :

TXTXX
XTXTX
XXTXT

total number of ways = 3*3! = 18

apart by two spaces :

TXXTX
XTXXT

total number of ways = 2*3! = 12

apart by three spaces
TXXXT

total number of ways = 3! = 6

total number of ways = 24+18+12+6 = 60.

how is the formula M!/L! is derived though?
_________________

for every person who doesn't try because he is
afraid of loosing , there is another person who
keeps making mistakes and succeeds..

Re: Combination   [#permalink] 20 Jun 2007, 17:25
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