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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

Answer E
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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

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)