johnwesley wrote:

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

A. 6

B. 30

C. 90

D. 120

E. 180

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.

Answer: E.

Hope it's clear.

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