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# How many different distinct ways can the letters in the word

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Senior Manager
Joined: 02 Oct 2005
Posts: 297
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How many different distinct ways can the letters in the word [#permalink]  02 Nov 2005, 08:45
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How many different distinct ways can the letters in the word
VACATION be arranged?

A. 25,375
B. 40,320
C. 52,500
D. 20,160
E. 5,040
Current Student
Joined: 28 Dec 2004
Posts: 3387
Location: New York City
Schools: Wharton'11 HBS'12
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Kudos [?]: 186 [0], given: 2

I get D also...

8!/2!...Cause letter A is repeated twice...
Current Student
Joined: 29 Jan 2005
Posts: 5240
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Kudos [?]: 191 [0], given: 0

Yep, D. 8!/2!=20160

How many different arrangements exist for the word ANTIDISESTABLISHMENTARIANISM?
Senior Manager
Joined: 02 Oct 2005
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Thats correct. OA is D and here goes the detailed explanation:

If you are arranging n items in a set, the number of different permutations possible is n!.

n! is pronounced n factorial.
n! = n(n-1)(n-2)(n-3) . . . * 2 * 1

For instance,
2! = 2 * 1
3! = 3 * 2 * 1
4! = 4 * 3 * 2 * 1

Since there are 8 letters in the word vacation, the number of ways to arrange the letters is 8! or 40320. However, this problem asks for the number of different distinct ways. Since there are two letter A's in the word, the different distinct ways of arranging the two A's are indistinguishable. To find the number of distinct permutations, divide the factorial of the elements in the set by the factorial of the number of identical elements.

Thus, the number of different distinct ways to arrange the letters in the word VACATION is (40320/2) or 20,160.
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Joined: 14 Oct 2003
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Re: PS: Probability - VACATION [#permalink]  02 Nov 2005, 19:40
Alright now how many ways to arrange the letters in the word chrysanthemum?
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Re: PS: Probability - VACATION   [#permalink] 02 Nov 2005, 19:40
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