Struggling with Permutation & Combination Basics

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eatpraylove: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Mon Jul 12, 2010 9:33 am

by eatpraylove » Fri Jul 25, 2014 3:23 am

This is a question from the Quants book Nova.
I am unable to solve it using the nPr formula. I have looked at the solution provided in the book but I want to crack it through the formula and not as "no. of ways to fill 1st box * no. of ways to fill 2nd box * 3rd.....4th)..which gives answer of 24. (4*3*2*1-24)".

The letters of the word JOHNY can be jumbled in 120 ways. In how many of them does the letter 'H'
appear in the middle?
(A) 1
(B) 20
(C) 24
(D) 26
(E) 30

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Source: Beat The GMAT — Problem Solving | Fri Jul 25, 2014 3:23 am

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by GMATinsight » Fri Jul 25, 2014 3:31 am

aakanksha003 wrote:
The letters of the word JOHNY can be jumbled in 120 ways. In how many of them does the letter 'H'
appear in the middle?
(A) 1
(B) 20
(C) 24
(D) 26
(E) 30

Let's fix the place of H at the middle of the number
_ _ H _ _

Now the remaining Four vacant places with 4 letters J, O, N and Y can be filled in

4 x 3 x 2 x 1 ways = 4! = 24 Ways

Answer: Option C

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Fri Jul 25, 2014 3:35 am

aakanksha003 wrote: The letters of the word JOHNY can be jumbled in 120 ways. In how many of them does the letter 'H'
appear in the middle?
(A) 1
(B) 20
(C) 24
(D) 26
(E) 30

ALTERNATE LOGICAL (PERHAPS THE BEST) METHOD:

The are 120 ways to arrange the letter [Given in the Question already]

Every letter has equal opportunity to occupy the same place out of 5 places therefore H will appear at every place out of 5 places = 120/5 = 24 times

Answer: Option C

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Fri Jul 25, 2014 3:44 am

aakanksha003 wrote:This is a question from the Quants book Nova.
I am unable to solve it using the nPr formula. I have looked at the solution provided in the book but I want to crack it through the formula and not as "no. of ways to fill 1st box * no. of ways to fill 2nd box * 3rd.....4th)..which gives answer of 24. (4*3*2*1-24)".

Hi Aakanksha,

Therea re several suggestion I want to convey:

1) A GMAT aspirant should not be very prone to be using formula otherwise somewhere you are definitely prone to stuck like you did here

2) The question which can be solved using "nPr" should preferably be solved using "nCr x r!" because nPr = nCr x r!. The suggestion is backed by the opinion that first you should be able to imagine that it requires selection using nCr first (if applicable) and then arrangement of selected r objects randomly

However, herr after fixing H at middle position you can arrange the 4 letter at 4 vacant places (2 towards the left of H and other 2 towards the right of H) in 4P4 ways

nPr = n!/(n-r)!

4P4 = 4!/((4-4)! = 4!/0! = 24/1 = 24

Answer: Option C

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by Brent@GMATPrepNow » Fri Jul 25, 2014 7:08 am

aakanksha003 wrote: The letters of the word JOHNY can be jumbled in 120 ways. In how many of them does the letter 'H' appear in the middle?
(A) 1
(B) 20
(C) 24
(D) 26
(E) 30

First of all, don't worry about permutations (i.e., the nPr formula). The truth of the matter is that true permutation questions are exceedingly rare on the GMAT. If we consult the Official Guide for GMAT Review - 13th edition (aka the OG13), we find a total of 5 counting questions, and not one of them is a TRUE permutation question.
As you might guess, I'm not a big fan of permutations. In fact, I even wrote an article about it: https://www.beatthegmat.com/mba/2013/07/ ... ons-part-i

You do, however, need to know how to apply the Fundamental Counting Principle (FCP). The FCP can be used to solve the majority of counting questions on the GMAT.

Here's how the FCP works in this question:

Take the task of arranging the 5 letters and break it into stages.

We'll begin with the most restrictive stage.

Stage 1: Select the middle letter
This letter must be H, so we can complete stage 1 in 1 way.

Stage 2: Select the 1st letter
There are now 4 letters remaining (J,O,N,Y), so we can complete this stage in 4 ways.

Stage 3: Select the 2nd letter
There are now 3 letters remaining, so we can complete this stage in 3 ways.

Stage 4: Select the 4th letter
There are now 2 letters remaining, so we can complete this stage in 2 ways.

Stage 5: Select the 5th letter
There is now 1 letter remaining, so we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus arrange the five letters) in (1)(4)(3)(2)(1) ways ([spoiler]= 24 ways[/spoiler])

Answer: C

----------------------------------
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