M14-25

Question

In how many ways can the letters of the word LEVEL be arranged such that the first letter is L and the last letter is E?

A. 4
B. 6
C. 10
D. 12
E. 24

Bunuel · Answer

Official Solution:

In how many ways can the letters of the word LEVEL be arranged such that the first letter is L and the last letter is E?

A. 4
B. 6
C. 10
D. 12
E. 24

We need to find all possible arrangements that fit the following pattern:  L---E . The three remaining distinct letters ( V ,  E , and  L ) can be arranged in the three slots between  L  and  E  in  3! = 6  different ways.

Answer: B

dsheth7 · Answer

How do we know not to count the arrangements of the alternate L and alternate E in the end spots? Is there verbiage that would have indicated otherwise if they did not count as the same? i.e.

My answer:

-----

First position: Choosing 1 L out of 2 L = 2 ways

Second, Third and Fourth : 3 Letters in 3! ways = 6

Last Position: 1 E out of 2 E's = 2 ways

Therefore, Total no of ways : 2*6*2 = 24.

Hence E

Kindly assist.

Bunuel · Answer

I think the best way to understand why this is wrong to list all possible cases. You'll see that there are only 6 cases, not 24:
LEVLE 
LELVE 
LVELE 
LVLEE 
LLEVE 
LLVEE

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Aishyk97 · Answer

Hi  Bunuel , I made the same mistake as  dsheth7 
Can you help with any similar problems to practice?

I think I got confused with problems including boys and girls.
If there are 3 boys and 2 girls to be seated in 5 chairs. A boy has to occupy the first chair and a girl has to occupy the last chari.

So then we would do 3C1 * 3! *2C1

Is that right? or is it still 3! itself?

Please help, thank you.

Bunuel · Answer

In scenarios involving distinguishable items, such as boys and girls, it's crucial to consider their arrangements since, for instance, the arrangement {Andrew, Bob} differs from {Bob, Andrew}. Therefore, if the question were, "In how many ways can three boys and two girls be arranged such that a boy occupies the first position and a girl occupies the last position?" the answer indeed would be 3C1 * 2C1 * 3!.

Here, 3C1 represents the number of ways to select a boy for the first position, 2C1 represents the number of ways to select a girl for the last position, and 3! represents the number of ways to arrange the remaining three children in between.

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Hope it helps.

Archit3110 · Answer

fix the position for L_ _ _ _ E 
left with 3 options in b/w EVL ; which can be arranged in 3! ways ; 6 option B

ritik0212 · Answer

Hi,

Why have we assumed the length to be 5 only?
When the words can be arranged to create words of length 4,3,2 as well. where we can keep L at the first position and E at last.

Bunuel · Answer

The problem specifically asks for the number of ways to  arrange  the letters of the word LEVEL, which means using all the existing letters of the word. The term "arrange" inherently implies forming permutations of the given letters without omitting any.

garvitpareek · Answer

so you have to divide your 24 by 2!*2! because there is 2L and 2E It will get 6.

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