Question

How many different arrangements can be made by using all the letters in the word ‘MATHEMATICS.’ How many of them begin with C? How many of them begin with T?

Answer 1

Given,

The word MATHEMATICS. 

It has 11 letters of which the letters M, A, and T are repeating and all letters are repeated twice.

Since we know, 

Permutation of n objects taking r at a time is ⁿP_r,and permutation of n objects taking all at a time is n!

And, 

We also know,

Permutation of n objects taking all at a time having p objects of same type, q objects of another type, r objects of another type is \(\frac{n!}{p!\times q!\times r!}\).

i.e, 

The number of repeated objects of same type are in denominator multiplication with factorial.

Total number of permutations of 11 objects with 3 objects repeating twice = \(\frac{11!}{2!\times 2!\times 2!}\)

= 4989600

To find the number of words starting with the letter C : 

This will be equal to permutation of 10 letters (excluded the letter C) where 3 letters (M, A, and T) are repeated twice = \(\frac{10!}{2!\times 2!\times 2!}\)

Total number of permutations of 11 objects with 3 objects repeating twice = \(\frac{11!}{2!\times 2!\times 2!}\)

= 4989600

To find the number of words starting with the letter C: This will be equal to permutation of 10 letters (excluded the letter C) where 3 letters (M, A, and T) are repeated twice = \(\frac{10!}{2!\times 2!\times 2!}\)

= 453600 

To find the number of words starting with the letter T : 

This will be equal to permutation of 10 letters (excluded the letter T) where 2 letters (M, A) are repeated twice, 

Which will be equals to :

⇒ \(\frac{10!}{2!\times 2!}\) 

= 907200

Hence,

A total number of words permuting the letters of word MATHEMATICS is 4989600. 

A total number of words starting with the letter C is 453600. 

A total number of words starting with the letter T equals 907200.