In how many ways can the 26 letters of the English alphabet be arrange

Question

Competition Mode Question

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 · Accepted Answer

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

freedom128 · Answer

(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)

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CareerGeek · Answer

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

exc4libur · Answer

TOTAL 26 LETTERS

A__ __B___ ___
The number ways to choose   in between is 24C7
The number of arrangements of   is 7!
The number of arrangements of A and B is 2
The number of arrangements of   and "A_ _B" as a group is 18!
Total arrangements 24!/7!17! * 7! * 2 * 18! = 36*24!

Ans (B)

eakabuah · Answer

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.

GMATGuruNY · Answer

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!

B

unraveled · Answer

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.

razorhell · Answer

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

aabhaas · Answer

A _ _ _ _ _ _ _ B(24C7, as A and B are fixed) + remaining letters(18!) 
+B _ _ _ _ _ _ _ A (24C7)+ remaining letters(18!)
--------------------------------------------------
=> 24C7 * 18! * 2
Ans.C

minustark · Answer

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.

Crytiocanalyst · Answer

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

bumpbot · Answer

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In how many ways can the 26 letters of the English alphabet be arrange

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