Probablity of picking up 3 vowels from the word "MATHEMATICS" Each of the letters in the word "MATHEMATICS" is on a letter tile in a
  bag. Foool picks three without replacement. what is the probability
  that he will get all vowels?


My approach,  

The number of ways to the letters of the words "MATHEMATICS" can be arranged 3 at a time is coefficient of $x^3 $ $$ 3! \times  (1+x)^5 \times \left(1+x+\frac{x^2}{2}\right)^3$$ which is $399$

The number of arrangements of 2A's, 1I and 1E taken 3 at at a time is coefficient of $x^3$ in  $$ 3! \times  (1+x)^2 \times  \left(1+x+\frac{x^2}{2}\right)$$ which is $12$.

Then the required probability is given by $\frac {12}{399} $ but apparently this is not the right answer. What exactly I am missing here?
How many ways are there to pick 3 letters? How many ways are there to pick 3 vowels? By my count, the right answer should be $$\binom{4}{3}/\binom{11}{3}=\frac{4}{165}.$$
We obtained  vowels in a word =5 (a ,e i,  o ,u ) 
    And numbers of word = 11

A/Q        the probability of picking up a vowel = 5/ 11  ans