All the letters of the word MATHEMATICS are arranged in all possible ways. Find the number of words in which there exists exactly 2 letters between the two strings “MAT” and “MAT”
Answers
Answer:
This is a simple yet interesting combinatorics problem.
First, let us find the total number of ways the 11 letters can be arranged.
Let f(x) represent the way that x letters can be arranged, where f(x)=x! .
This is because if there are x places for the letters to be placed, the first spot can have x , the second x−1 , all the way until the x th spot can have only 1 possible value.
x(x−1)...1=x!.
There are 11 letters in the word “mathematics”, so we find f(11) .
f(11)=11! .
Using f(11) would suffice if all 11 letters in the word were distinct. However, since there are repetitions of letters, and each of those same letters are not distinct (e.g. the word “mathematics” is unchanged even if the two ‘a’s are swapped), we must divide f(11) by f(n) , where n is the number of times each letter shows up.
Note that for letters that only show up once in the word,
f(n)=f(1)=1!=1.
Therefore, we do not need to include these letters in our final division, but only for letters that show up more than once.
In the case of the word “mathematics”, the letters “a”, “m”, and “t” show up twice.
Therefore, the number of ways that the letters of the word “mathematics” can be arranged is:
f(11)f(2)⋅f(2)⋅f(2)
=11!2!2!2!
=4989600
Therefore, the letters in “mathematics” can be arranged in 4,989,600 ways.