Answer to Question #123075 in Combinatorics | Number Theory for Hussain Ousman Darboe

Question

Find the number of permutations of four letters from the word MATHEMATICIAN

Answer 1

We have two "M,three A,two T,two I and H,E,N are distinct.

Case 1: words with distinct letters .

We have M,A,T,I,H,E,N ,seven distinct letter , so 7P4 *4!=20160 ways

Case 2: Words with exactly a letter repeated twice.

We have M, A,T, I repeating itself. Now one of this three letter can be choose in 4C1= 4 ways The other two distinct letters can be selected in 6C2= 15 ways

Now each combination can be arranged in= 4!/2!=12 ways

So total number of such words=4*15*12= 720 ways

Case 3: Words with exactly two distinct letters repeated twice

Two letters out of the four repeating letters M,A,T,I can be selected in 4C2 =6 ways .Now each combination can be arranged in 4!/(2!*2!)=6 ways .

So, total number of such words=6*6=36 ways.

Case 4:Words with exactly a letter repeated thrice

We have one portion for this as our main letter that is A.Now we have to select 1 letter out of the 6 remaining options so number of ways to select this 6C1=6 ways.Now each combination can be arranged in 4!/3!=4 ways.

So, total number of such words=6*4=24 ways

So, all possible number of arrangements= 20160+720+36+24=20940 ways.