Problem Solving

Gmat_mission wrote: ↑

Wed Jun 24, 2020 7:51 am

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

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

[spoiler]OA=E[/spoiler]

Source: GMAT Paper Tests

------ASIDE-----------------------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]

So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are 11 letters in total
There are 4 identical I's
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]
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Now on to the question!

The word: COMMON:
There are 6 letters in total
There are 2 identical O's
There are 2 identical M's
So, the total number of possible arrangements = 6!/[(2!)(2!)] = 180

Answer: E

Cheers,
Brent

Gmat_mission wrote: ↑

Wed Jun 24, 2020 7:51 am

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

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

[spoiler]OA=E[/spoiler]

Source: GMAT Paper Tests

Solution:

We use the indistinguishable permutations formula to account for the two identical C’s and two identical O’s in the word. Thus, the number of ways to arrange the letters in COMMON is:

6! / (2! x 2!) = (6 x 5 x 4 x 3 x 2) / (2 x 2) = 6 x 5 x 3 x 2 = 180

If the letters were all distinct, the answer would be 6!. However, there are two O’s and two M’s, and so we divide 6! by 2! x 2! to take into account the permutations that are not distinct due to the identical O’s and M’s in the word COMMON.

Answer: E