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

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Math Expert
Joined: 02 Sep 2009
Posts: 52344
In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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08 Apr 2016, 02:13
12
00:00

Difficulty:

55% (hard)

Question Stats:

52% (01:20) correct 48% (01:35) wrong based on 125 sessions

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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

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Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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08 Apr 2016, 06:40
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

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Joined: 05 Feb 2016
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GMAT 1: 510 Q39 V21
Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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09 Apr 2016, 01:29
1
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
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Joined: 26 Mar 2017
Posts: 1
Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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17 Apr 2017, 13:33
Hello,

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

Thank You
CEO
Joined: 11 Sep 2015
Posts: 3355
Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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17 Apr 2017, 14:59
1
Top Contributor
Bunuel wrote:
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

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)

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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Joined: 11 Sep 2015
Posts: 3355
Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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17 Apr 2017, 14:59
Top Contributor
praveen sreedhar wrote:
Hello,

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

Thank You

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

Cheers,
Brent
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Re: In how many ways can the letters of the word SPENCER be arranged if th  [#permalink]

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05 May 2018, 11:12
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Re: In how many ways can the letters of the word SPENCER be arranged if th &nbs [#permalink] 05 May 2018, 11:12
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