M37-44

Question

In how many ways can the letters of the word PERMUTATIONS be arranged such that the order of vowels remains unchanged?

A.  \frac{12!}{2!} 
B.  \frac{12!}{5!} 
C.  \frac{12!}{2!*5!} 
D.  \frac{4*7!}{2!} 
E.  \frac{4*7!}{2!*5!}

hr1212 · Accepted Answer

Another way to think could be,

Total possible arrangements = 12!/2! (T is repeated twice)

Now notice that there are 5 different vowels and we want only one order out of the default 5! orders, so we will need to remove rest of the orders but one, which is nothing but,

12!/2! * 1/5! = 12!/(2!*5!)

It's like saying for ABC, out of all the orders possible, I just need ABC ie. 1 out of 3!

Ans: C

Bunuel · Answer

Official Solution:

In how many ways can the letters of the word PERMUTATIONS be arranged such that the order of vowels remains unchanged?

A.  \frac{12!}{2!} 
B.  \frac{12!}{5!} 
C.  \frac{12!}{2!*5!} 
D.  \frac{4*7!}{2!} 
E.  \frac{4*7!}{2!*5!}

PERMUTATIONS has 12 letters: 5 vowels (EUAIO) and 7 consonants (PRMTTNS).
 
Choose 5 slots from 12 for the vowels:  C^5_{12}=\frac{12!}{5!*7!} . Since the order of the vowels must remain unchanged (EUAIO), then we do not arrange them.
 
7 consonants in 7 remaining slots can be arranged in  \frac{7!}{2!}  ways (notice that we have two T's among 7 consonants).
 
So, the total number of ways is  \frac{12!}{5!*7!}*\frac{7!}{2!}=\frac{12!}{2!*5!} .

Answer: C

agrasan · Answer

I like the solution - it’s helpful.

Nipunh · Answer

I like the solution - it’s helpful.

siddharth_ · Answer

Another way to think could be -- 
remove the arrangements for vowels like for Ts. 12!/2!5!

patnaik1988 · Answer

I like the solution - it’s helpful.

akkkk1 · Answer

Hi, 
I got the same answer but using a separate methodology (I was helped by the options to an extent).

Ways to arrange all = 12!
Divide by 5! because we do not want to arrange the 5 vowels.
Divide by 2! because we do not want to arrange the 2 T's.

Is this reasoning correct?

Thanks

Bunuel · Answer

Yes, the method is correct, but the reasoning should be stated more precisely. The number of ways to arrange the word PERMUTATIONS is 12!/2! because the two T's are identical. Then we multiply by 1/5! because, among the 5! possible relative orders of the vowels, only one, EUAIO, is allowed.

Thus, the count is 12!/2! * 1/5!.

Thus, the count is 12!/2! * 1/5!.

akkkk1 · Answer

Thank you!

M37-44

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