So I took the vowels (a,e,i) permute them 3p3 and then I choose 6c2.Here I get baffled and can't figure out what should I do next.
By the way multiply 3p3 and 6c2 comes up with 180 which is not the answer.The answer is 1800.
Please explain where I made the mistake.Or Am I missing something?
$\endgroup$3 Answers
$\begingroup$You need to first choose the 5 letters in the word, then permute it.
You must choose all 3 vowels, and 2 consonants from the 6; there are $6C2 = 15$ choices.
Then you permute the 5 chosen letters; there are $5P5 = 5!$ choices.
Multiplying, we have $6C2 \times 5! = 15 \times 120 = 1800$ different words.
$\endgroup$ 3 $\begingroup$You need to choose $2$ out of the $6$ letters. That is ${6 \choose 2}$ ways as you said.
Now you have $5$ letters and so just permute them. So your answer is ${6 \choose 2} \times 5!$
$\endgroup$ $\begingroup$Firstly Take 3 vowels (1 way) & take 2 consonants out of 6 consonants => 6C2 = 1 * 15 = 15 ways to select required 5 letters.
Secondly, the number of ways to arrange those 5 letters is 54321 = 120
The answer is 120 * 15 = 1800
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