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In how many ways can the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B?
A. \(12\)
B. \(36 * 24!\)
C. \(24C7 * 18! * 2\)
D. \(26C7 × 20! × 2\)
E. \(26P7 × 20! × 2\)
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Kinshook
Major Poster
Joined: 03 Jun 2019
Last visit: 11 Sep 2026
Posts: 6,127
Location: India
Schools: HBS '28 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
In how many ways can the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B?
A. 12
B. 36∗24!
C. 24C7∗18!∗2
D. 26C7×20!×2
E. 26P7×20!×2
Between A & B, there are 7 letters out of 24.
Number of ways of selecting 7 letters out of 24 = 24C7
Arranging these 7 numbers between A & B = 7!
Number of ways of arranging A & B = 2
Number of ways 9 letter word with A & B at both ends = 24C7*7!*2
Consider this 9 letter word as a single letter
Number of ways of arranging the 18 letters = 18!
Total number of ways the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B = 24C7*7!*2*18! = (24!/7!17!)*7!*2*18! = 24! *2*18 = 36*24!
IMO B
A. 12
B. 36∗24!
C. 24C7∗18!∗2
D. 26C7×20!×2
E. 26P7×20!×2
Between A & B, there are 7 letters out of 24.
Number of ways of selecting 7 letters out of 24 = 24C7
Arranging these 7 numbers between A & B = 7!
Number of ways of arranging A & B = 2
Number of ways 9 letter word with A & B at both ends = 24C7*7!*2
Consider this 9 letter word as a single letter
Number of ways of arranging the 18 letters = 18!
Total number of ways the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B = 24C7*7!*2*18! = (24!/7!17!)*7!*2*18! = 24! *2*18 = 36*24!
IMO B
General Discussion
GMAT 1: 720 Q49 V40
Posts: 939
(1) Scheme A,_,_,_,_,_,_,_,B,(remaining)
A can be positioned from letters 1 to 18. Accordingly, B is positioned from 9 to 26. The remaining 24 letters can fill the spaces with no repetition allowed in 24! ways.
----> No of ways = 18*24!
(2) Similarly, the reverse scheme B,_,_,_,_,_,_,_,A,(remaining) is also applicable
----> Additional no of ways = 18*24!
Total no of ways = 18*24! + 18*24! = 36*24!
FINAL ANSWER IS (B)
Posted from my mobile device
A can be positioned from letters 1 to 18. Accordingly, B is positioned from 9 to 26. The remaining 24 letters can fill the spaces with no repetition allowed in 24! ways.
----> No of ways = 18*24!
(2) Similarly, the reverse scheme B,_,_,_,_,_,_,_,A,(remaining) is also applicable
----> Additional no of ways = 18*24!
Total no of ways = 18*24! + 18*24! = 36*24!
FINAL ANSWER IS (B)
Posted from my mobile device
CareerGeek
Joined: 20 Jul 2017
Last visit: 11 Sep 2026
Posts: 1,285
Location: India
Concentration: Entrepreneurship, Marketing
GMAT 1: 690 Q51 V30
WE:Education (Education)
GMAT 1: 690 Q51 V30
Posts: 1,285
Updated on: 20 Mar 2020, 01:15
Originally posted by CareerGeek on 20 Mar 2020, 01:07.
Last edited by CareerGeek on 20 Mar 2020, 01:15, edited 1 time in total.
Last edited by CareerGeek on 20 Mar 2020, 01:15, edited 1 time in total.
A & B have to be separated by 7 letters
Make a block of 9 letters with {A/B, _, _, _, _, _, _, _, B/A} and remaining 17 letters
We can arrange 7 letters from 24 available in \(24p_7\) ways
A & B can be interchanged in 2 ways
18 units (1 block + remaining 17 letters) can be arranged in 18! ways
--> Total number of ways = 2*24p7*18! = 2*{24*23*22*21*20*19*18}*18! = 2*24!*18 = 36*24!
Option B
Make a block of 9 letters with {A/B, _, _, _, _, _, _, _, B/A} and remaining 17 letters
We can arrange 7 letters from 24 available in \(24p_7\) ways
A & B can be interchanged in 2 ways
18 units (1 block + remaining 17 letters) can be arranged in 18! ways
--> Total number of ways = 2*24p7*18! = 2*{24*23*22*21*20*19*18}*18! = 2*24!*18 = 36*24!
Option B
Quote:
TOTAL 26 LETTERS
A__[7]__B___[17]___
The number ways to choose [7] in between is 24C7
The number of arrangements of [7] is 7!
The number of arrangements of A and B is 2
The number of arrangements of [17] and "A_[7]_B" as a group is 18!
Total arrangements 24!/7!17! * 7! * 2 * 18! = 36*24!
Ans (B)
If we take out A and B from the 26 letters, we are left with 24 letters.
The number of ways that 7 letters can be selected out of 24 = 24C7.
The number of ways each of the 7 letters selected can be arranged = 7!
After selecting 7 letters out of 24, the number of letters left = 24-7=17
In how many ways can we arrange the remaining 17 letters? 17!
Each of the nine letters formed by interspersing 7 letters between letters A and B can be arranged between the 17 remaining letters in 17+1=18 ways
Last but not the least, the two letters A and B can be arranged interchangeably at the ends in 2 ways
Total No. of ways = \(24C7*7!*17!*18*2\) = \((24!/(7!*17!))\)\(*7!*17!*18*2\) = \(36*24!\)
The answer is B.
The number of ways that 7 letters can be selected out of 24 = 24C7.
The number of ways each of the 7 letters selected can be arranged = 7!
After selecting 7 letters out of 24, the number of letters left = 24-7=17
In how many ways can we arrange the remaining 17 letters? 17!
Each of the nine letters formed by interspersing 7 letters between letters A and B can be arranged between the 17 remaining letters in 17+1=18 ways
Last but not the least, the two letters A and B can be arranged interchangeably at the ends in 2 ways
Total No. of ways = \(24C7*7!*17!*18*2\) = \((24!/(7!*17!))\)\(*7!*17!*18*2\) = \(36*24!\)
The answer is B.
Posts: 1,348
7 letters must separate A and B.
Implication:
In the 26-letter arrangement, A and B must occupy the ends of a 9-letter block, as in the following examples:
A=1 __ __ __ __ __ __ __ B=9
A=2 __ __ __ __ __ __ __ A=10
A=17 __ __ __ __ __ __ __ B=25
A=18 __ __ __ __ __ __ __ B=26
The value in red indicates that the number of ways to position this 9-letter block = 18
Number of ways to arrange A and B at the ends of this block = 2!
Number of ways to arrange the remaining 24 letters = 24!
To combine these options, we multiply:
18 * 2! * 24! = 36 * 24!
_________________
Implication:
In the 26-letter arrangement, A and B must occupy the ends of a 9-letter block, as in the following examples:
A=1 __ __ __ __ __ __ __ B=9
A=2 __ __ __ __ __ __ __ A=10
A=17 __ __ __ __ __ __ __ B=25
A=18 __ __ __ __ __ __ __ B=26
The value in red indicates that the number of ways to position this 9-letter block = 18
Number of ways to arrange A and B at the ends of this block = 2!
Number of ways to arrange the remaining 24 letters = 24!
To combine these options, we multiply:
18 * 2! * 24! = 36 * 24!
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In how many ways can the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B?
A. 12
B. 36∗24!
C. 24C7∗18!∗2
D. 26C7×20!×2
E. 26P7×20!×2
Assumption: Arrangements is linear not circular.
A + seven alphabets + B = 9 alphabets.
A and B can be written in 2 ways:
1. AB
2. BA
Rest of the places can be filled with any random alphabets that are left i.e. 24 alphabets.
A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (each space is an alphabet)
_ A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
...
...
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ A _ _ _ _ _ _ _ B
Since either of A or B can take place till 9th last alphabet i.e. 18th position,
total positions that A or B can take = 18
So, Total ways of arrangements =
24! * 18 * 2 = 24! * 36
Answer B.
_________________
A. 12
B. 36∗24!
C. 24C7∗18!∗2
D. 26C7×20!×2
E. 26P7×20!×2
Assumption: Arrangements is linear not circular.
A + seven alphabets + B = 9 alphabets.
A and B can be written in 2 ways:
1. AB
2. BA
Rest of the places can be filled with any random alphabets that are left i.e. 24 alphabets.
A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (each space is an alphabet)
_ A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ A_ _ _ _ _ _ _ B _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
...
...
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ A _ _ _ _ _ _ _ B
Since either of A or B can take place till 9th last alphabet i.e. 18th position,
total positions that A or B can take = 18
So, Total ways of arrangements =
24! * 18 * 2 = 24! * 36
Answer B.
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razorhell
Joined: 14 Jul 2017
Last visit: 26 Jun 2024
Posts: 74
Location: India
Schools: Booth '24 Darden '24 Fuqua '24 ISB '23 Kellogg '24 LBS '24 NTU NUS Ross '24 Rotman '24 Tuck '24 Columbia CBS '24 Johnson '24 Wharton '24
GMAT 1: 680 Q49 V35
GPA: 3.76
WE:Management Consulting (Consulting)
Schools: Booth '24 Darden '24 Fuqua '24 ISB '23 Kellogg '24 LBS '24 NTU NUS Ross '24 Rotman '24 Tuck '24 Columbia CBS '24 Johnson '24 Wharton '24
GMAT 1: 680 Q49 V35
Posts: 74
26 - 9 = 17 ... If we consider the 9 letters as 1 objects we have to permute 18 different objects .. b and A can interchange ... I think the answer is C
A _ _ _ _ _ _ _ B(24C7, as A and B are fixed) + remaining letters(18!)
+B _ _ _ _ _ _ _ A (24C7)+ remaining letters(18!)
--------------------------------------------------
=> 24C7 * 18! * 2
Ans.C
+B _ _ _ _ _ _ _ A (24C7)+ remaining letters(18!)
--------------------------------------------------
=> 24C7 * 18! * 2
Ans.C
minustark
Joined: 14 Jul 2019
Last visit: 01 Apr 2021
Posts: 465
Status:Student
Location: United States
Concentration: Accounting, Finance
Schools: Zicklin Baruch "22
GMAT 1: 650 Q45 V35
GPA: 3.9
WE:Education (Accounting)
Products:
In how many ways can the 26 letters of the English alphabet be arranged so that there are seven letters between the letters A and B?
The position of A, B and the other 7 letters can be fixed in the following ways :
A 24C7 B
B 24C7 A
So, there can be 2 * 24C7 arrangements for these 9 letters. Now we can say these 9 letters are serving as a single letter. So the total letters become 18. These 18 letters can be arranged in 18! ways. So, the total arrangements will become : 2* 24C7 * 18!.
C is the answer.
The position of A, B and the other 7 letters can be fixed in the following ways :
A 24C7 B
B 24C7 A
So, there can be 2 * 24C7 arrangements for these 9 letters. Now we can say these 9 letters are serving as a single letter. So the total letters become 18. These 18 letters can be arranged in 18! ways. So, the total arrangements will become : 2* 24C7 * 18!.
C is the answer.
Posts: 938
Arranging these 7 numbers between A & B = 24*23*.....18
Number of ways of arranging A & B = 2!
Considering all the 9 leeters as one arrangement
Number of ways of arranging the 18 letters = 18!
Therefore the tolal way of arranging the letters equal= 24*23....18*18!*2
=>24! *2*18 = 36*24
Therefore IMO B
Number of ways of arranging A & B = 2!
Considering all the 9 leeters as one arrangement
Number of ways of arranging the 18 letters = 18!
Therefore the tolal way of arranging the letters equal= 24*23....18*18!*2
=>24! *2*18 = 36*24
Therefore IMO B
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