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.
(b) Determine the probability that the vowels are placed together.
~~~~~~~~~~~~~~~~~~~~~


<pre>
(a)  The word MATHEMATICS has 11 letter.

     Of them, letters M, A, T are repeating; the other 8 = 11-3 letters are unique.

     Letter M has multiplicity 2; the same with letters A and T.


     Therefore, the number of different arrangements of the letters of word MATHEMATICS is

         {{{11!/(2!*2!*2!)}}} = {{{(1*2*3*4*5*6*7*8*9*10*11)/(2*2*2)}}} = 4,989,600.    <U>ANSWER</U>



(b)  The block of vowels is AEAI.  It consists of 4 letters.

     The other 11-4 = 7 letters are consonants.

     When we consider arrangements, the block of vowels can be placed in any of 7+1 = 8 
     possible positions between consonants, or before consonants, or after consonants.
     It gives 8 possible options.

     Next, there are 4! = 24 possible permutations inside this block,
     and there are 7! possible permutations of consonants outside of this block.


     So, the total number of such permutations for word MATHEMATICS is the product
             8*4!*7!

     To get the number of possible arrangements, we still (or again) must divide it by the factor (2!*2!*2!).


     So, the number of all possible arrangements of this type (b) is  

          {{{(8*4!*7!)/(2!*2!*2!)}}} = {{{(8*24*1*2*3*4*5*6*7)/8}}} = 120,960.    


     The desired probability is the ratio of numbers  120,960  and  4,989,600

         P = {{{120960/4989600}}} = {{{4/165}}} = 0.02424  (rounded).    <U>ANSWER</U>
</pre>

Solved.