# Thread: How many permutations are there?

1. ## How many permutations are there?

The English alphabet has 26 letters, 5 of these are vowels.
How many permutations of English letters are there in which all vowels are in consecutive places?

My answer:
There are C(26,5) + C(26,26-5) = C(26,5) + C(26,21) = 2·C(26,5) = 2·65780 = 131560 permutations.

Is my answer correct? If not, please explain how one should address this problem. Thanks.

2. Originally Posted by feyomi
The English alphabet has 26 letters, 5 of these are vowels.
How many permutations of English letters are there in which all vowels are in consecutive places?

My answer:
There are C(26,5) + C(26,26-5) = C(26,5) + C(26,21) = 2·C(26,5) = 2·65780 = 131560 permutations.
No that is not correct.
Here is the answer in the back of the book: $\displaystyle (5!)(22!)$.
You tell us why.

3. Originally Posted by Plato
No that is not correct.
Here is the answer in the back of the book: $\displaystyle (5!)(22!)$.
You tell us why.

Ah right okay, so there are 5! ways of arranging the vowels and 21! ways of arranging the consonants, and by rule of product, the answer is 5!21! ?

Did you mean 21 instead of 22? If not, I'm not sure where the 22 comes from

4. Originally Posted by feyomi
Ah right okay, so there are 5! ways of arranging the vowels and 21! ways of arranging the consonants, and by rule of product, the answer is 5!21! ?

Did you mean 21 instead of 22? If not, I'm not sure where the 22 comes from
The five vowels form one block. With the twenty-one consonants that gives us 22 blocks to arrange.

### how many permutations of english letters are there in which all vowels are in consecutive places

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