In how many ways can the letters of the word SPENCER be arranged if th

Question

In how many ways can the letters of the word SPENCER be arranged if the S and P must always be together and the N and C must always be together?

A. 12 
B. 24 
C. 60 
D. 120 
E. 240

BrentGMATPrepNow · Accepted Answer

Take the task of arranging the letters and break it into   stages  .

Stage 1 : Glue S and P together. 
Note, this will ensure that S and P are together. 
There are 2 ways to glue S and P together: SP and PS
So we can complete stage 1 in  2  ways

Stage 2 : Glue N and C together. 
There are 2 ways to glue N and C together: NC and CN
So we can complete stage 2 in  2  ways

IMPORTANT: We now have 5 "objects" to arrange. They are: E, E, R, S/P combo, N/C combo

Stage 3 : arrange the 5 "objects" in a row

--------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 !)]
---------BACK TO THE QUESTION--------------------

In stage 3, we must arrange 5 objects: E, E, R, S/P combo, N/C combo
There are  5  objects in total 
There are  2  identical E's
So, the total number of possible arrangements =  5 !/ 2 !)] =  60

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all of the letters) in  (2)(2)(60)  ways (= 240 ways)

Answer:  E

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Skywalker18 · Answer

There are 7 letters in the word SPENCER are 2E , SP , NC , R

We will consider SP and NC as a group and there are 2 E's

No of arrangements in which S and P must always be together and the N and C must always be together
= (5! *2 *2)/2!
= 5!*2 
= 120 *2
= 240

Answer E

Abkhazian · Answer

SPENCER

1 x SP (or PS)
2 x E's
1 x NC (or CN)
1 x R
-----------
5 entities where two are alike / and two have 2 possibilities

# of ways = (5!)(2)(2) / (2!) = 240

praveen sreedhar · Answer

Hello,

Could you please explain the solution for this problem using the counting principle in detail

Thank You

BrentGMATPrepNow · Answer

Above, you'll find a solution that uses the Fundamental Counting Principle (FCP)

Cheers, 
Brent

bumpbot · Answer

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In how many ways can the letters of the word SPENCER be arranged if th

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