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QOTD#19 The 5 letters in the list G, H, I, J, K are to be

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QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 12 Sep 2016, 12:16
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The 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are there ?

A. 60
B. 36
C. 24
D. 12
E. 6

Practice Questions
Question: 14
Page: 206
[Reveal] Spoiler: OA

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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 12 Sep 2016, 12:18
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Explanation

When the 5 letters are rearranged, G is to be listed in the 3rd position and H cannot be next to G, so there are only two possible positions for H: 1st and 5th.

Case 1: In the rearranged list, G is in the 3rd position, H is in the 1st position, and each of the remaining 3 letters can be in any of the remaining 3 positions. The number of ways these remaining 3 letters can be arranged is 3!, or 6. Thus the total number of rearrangements in case 1 is 6.

Case 2: In the rearranged list, G is in the 3rd position, H is in the 5th position, and each of the remaining 3 letters can be in any of the remaining 3 positions. Thus, as in case 1, the total number of rearrangements in case 2 is 6.

Thus there are 6 + 6, or 12, possible rearrangements, and the correct answer is Choice D.
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 20 Sep 2016, 05:53
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sandy wrote:
The 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are

A. 60
B. 36
C. 24
D. 12
E. 6


Another approach:
Take the task of arranging the 7 digits and break it into stages.

We’ll begin with the most restrictive stages.

Stage 1: Place the letter G
Since the letter G must be placed in the 3rd position, we can complete stage 1 in 1 way

Stage 2: Place the letter H
So far we have: _ _ G _ _
Since H cannot be next to G, we can only place H in the 1st position or the 5th position
So we can complete this stage in 2 ways.

Stage 3: Place the letter I
We can place I wherever we wish.
There are 3 spaces remaining, so we can complete this stage in 3 ways.

Stage 4: Place the letter J
We can place I wherever we wish.
There are 2 spaces remaining, so we can complete this stage in 2 ways.

Stage 5: Place the letter K
There is 1 space 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 all 5 letters) in (1)(2)(3)(2)(1) ways (= 12 ways)

Answer: D

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Brent
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 27 Jul 2018, 02:17
sandy wrote:
Explanation

When the 5 letters are rearranged, G is to be listed in the 3rd position and H cannot be next to G, so there are only two possible positions for H: 1st and 5th.

Case 1: In the rearranged list, G is in the 3rd position, H is in the 1st position, and each of the remaining 3 letters can be in any of the remaining 3 positions. The number of ways these remaining 3 letters can be arranged is 3!, or 6. Thus the total number of rearrangements in case 1 is 6.

Case 2: In the rearranged list, G is in the 3rd position, H is in the 5th position, and each of the remaining 3 letters can be in any of the remaining 3 positions. Thus, as in case 1, the total number of rearrangements in case 2 is 6.

Thus there are 6 + 6, or 12, possible rearrangements, and the correct answer is Choice D.

@sandy why can't we do like this
1] total no of arrangements 5!=120
2] then place H in 2nd and 4th position, so no of ways are 12
3] then subtract 120-12 = 108

why is this approach wrong?
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 27 Jul 2018, 06:53
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ssp4all wrote:
@sandy why can't we do like this
1] total no of arrangements 5!=120
2] then place H in 2nd and 4th position, so no of ways are 12
3] then subtract 120-12 = 108

why is this approach wrong?


G is not in the third place in your assumption. If G is in third place

1. total number of arrangements 4!=24 (Not 5! as G is fixed to 3rd place)
2] then place H in 2nd and 4th position, so no of ways are 12
3] then subtract 24-12 = 12

Now its OK
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 18 Oct 2019, 02:09
GreenlightTestPrep wrote:
sandy wrote:
The 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are

A. 60
B. 36
C. 24
D. 12
E. 6


Another approach:
Take the task of arranging the 7 digits and break it into stages.

We’ll begin with the most restrictive stages.

Stage 1: Place the letter G
Since the letter G must be placed in the 3rd position, we can complete stage 1 in 1 way

Stage 2: Place the letter H
So far we have: _ _ G _ _
Since H cannot be next to G, we can only place H in the 1st position or the 5th position
So we can complete this stage in 2 ways.

Stage 3: Place the letter I
We can place I wherever we wish.
There are 3 spaces remaining, so we can complete this stage in 3 ways.

Stage 4: Place the letter J
We can place I wherever we wish.
There are 2 spaces remaining, so we can complete this stage in 2 ways.

Stage 5: Place the letter K
There is 1 space 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 all 5 letters) in (1)(2)(3)(2)(1) ways (= 12 ways)

Answer: D

Cheers,
Brent

I liked ur explanation.
My question:

if we have to put G in the 3rd then why couldn't we solve it like this : 4 2 1 1 1

the first blank I can use H I J K, the second I can use any of ( J I K ), then G , then any of the rest except H, then the remaining.
what is wrong with that answer?
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 18 Oct 2019, 03:18
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sandy wrote:
The 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are there ?

A. 60
B. 36
C. 24
D. 12
E. 6

Practice Questions
Question: 14
Page: 206



total 12 ways(6 plus 6) with first way being placing the at first and in another way one can place H at the last.
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The 5 letters in the list G, H, I, J, K are to be rearranged [#permalink] New post 11 Nov 2019, 00:59
The 5 letters in the list G, H, I, J, K are to be rearranged so that G is the 3rd letter in the list and H is not next to G. How many such rearrangements are there?

A. 60
B. 36
C. 24
D. 12
E. 6
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Re: The 5 letters in the list G, H, I, J, K are to be rearranged [#permalink] New post 11 Nov 2019, 01:54
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There are total 5 slots for the 5 letters.
- - - - -
G has to go in the 3rd slot
- - 1 - - because there is only 1 option i.e. 'G' for the 3rd slot
H cannot be on either side of G that leaves us with 3 letters I J K
- 3 1 2 - Among those 3 letters we chose 1 and that leaves us with 2 other letters.
After spot for three of the letters are finalized we have two letters left hence
2 3 1 2 1 = 12. Remember this is a permutation problem so order matters
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Re: The 5 letters in the list G, H, I, J, K are to be rearranged [#permalink] New post 12 Nov 2019, 10:51
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[

I'd typically solve this question the same way amorphous did.
However, since the answer choices are relatively small, it's not a crazy idea to consider LISTING AND COUNTING all possible outcomes (especially if you're not sure how to solve the question.

If we're take a systematic approach, it won't take long.
We get:

HIGJK
HIGKJ
HJGIK
HJGKI
HKGIJ
HKGJI
IJGKH
IKGJH
JIGKH
JKGIH
KIGJH
KJGIH

TOTAL = 12

Answer: D

Cheers,
Brent
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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be [#permalink] New post 16 Nov 2019, 12:28
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Merged similar topics.

Brent also explained it two times.

Use the search button before to post a question https://greprepclub.com/forum/search.php

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Re: QOTD#19 The 5 letters in the list G, H, I, J, K are to be   [#permalink] 16 Nov 2019, 12:28
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