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# How many ways can the letters in the word COMMON be arranged

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Manager
Joined: 24 Jan 2013
Posts: 72
How many ways can the letters in the word COMMON be arranged  [#permalink]

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22 Feb 2013, 07:21
1
5
00:00

Difficulty:

5% (low)

Question Stats:

77% (00:46) correct 23% (01:27) wrong based on 159 sessions

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How many ways can the letters in the word COMMON be arranged?

A. 6
B. 30
C. 90
D. 120
E. 180

Spoiler: :: How to solve
This is a permutation with indistinguishable events - repeated items. The number of different permutations of N objects, where there are N1 indistinguishable objects of style 1, N2 indistinguishable objects of style 2, ..., and Nk indistinguishable objects of style k, is = N!/(N1!*N2!* ... * Nk!). In this case, N=6; N1=2, and N2=2. This gives the formula: 6!/(2!*2!)=180
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: How many ways can the letters in the word COMMON be arranged  [#permalink]

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22 Feb 2013, 07:28
6
4
johnwesley wrote:
How many ways can the letters in the word COMMON be arranged?

A. 6
B. 30
C. 90
D. 120
E. 180

Spoiler: :: How to solve
This is a permutation with indistinguishable events - repeated items. The number of different permutations of N objects, where there are N1 indistinguishable objects of style 1, N2 indistinguishable objects of style 2, ..., and Nk indistinguishable objects of style k, is = N!/(N1!*N2!* ... * Nk!). In this case, N=6; N1=2, and N2=2. This gives the formula: 6!/(2!*2!)=180

THEORY FOR SUCH KIND OF PERMUTATION QUESTIONS:

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is $$8!$$ as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

BACK TO THE ORIGINAL QUESTION:
How many ways can the letters in the word COMMON be arranged?
A. 6
B. 30
C. 90
D. 120
E. 180

According to the above the # of permutations of 6 letters COMMON out of which 2 O's and 2 M's are identical is $$\frac{6!}{2!*2!}=180$$.

Hope it's clear.
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Re: How many ways can the letters in the word COMMON be arranged  [#permalink]

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06 Feb 2019, 05:40
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Re: How many ways can the letters in the word COMMON be arranged   [#permalink] 06 Feb 2019, 05:40
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