Question 1203283
The letters of the word MATHEMATICS are written, one on each of 11 separate
cards. The cards are laid out in a line.
a. Calculate the number of different arrangements of these letters.<pre>
If all 11 letters of MATHEMATICS were distinguishable, the answer would be 11!.
But since the two M's are indistinguishable we must divide by 2!. Same for the A's and T's.

So that's {{{11!/(2!2!2!)}}}{{{""=""}}}{{{4989600)}}}</pre>
b. Determine the probability that the vowels are placed together.<pre>

We begin by finding the permutations of consonants MTHMTCS.
There are {{{7!/(2!2!)}}}{{{""=""}}}{{{1260}}}

Then for every arrangement of consonants, for instance for MTHMTCS, we place 8
blanks.

____M____T____H____M____T____C____S____

and pick one of the 8 blanks to put all 4 vowels together in.  We can do that in
8 ways.

The 4 vowels AEAI can be arranged in {{{4!/2!}}}{{{""=""}}}{{{12}}} ways.

So the number of successful ways is (1260)(8)(12) = 120960 ways.

So the desired probability is {{{120960/4989600}}}{{{""=""}}}{{{4/165}}}

Edwin</pre>