# 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 $$\binom{4}{3}/\binom{11}{3}=\frac{4}{165}.$$

-
Thanks Arturo, I just over-complicated a simple problem :( –  Quixotic Mar 4 '12 at 8:36
You have four items: four physical tiles; they are different objects, because they are different tiles. Just because two of them have the same letter written on them does not make them the same tile! –  Arturo Magidin Mar 4 '12 at 8:40
Thanks again Arturo, I was all messed up. –  Quixotic Mar 4 '12 at 8:42
I don't know what you're missing, because I don't know why you're taking the approach you take. The answer is $(4/11)(3/10)(2/9)$.