How many different arrangements can be made by using all the letters in the word ‘MATHEMATICS’.
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?

Solution:

The word MATHEMATICS consists of 11 letters that include two Ms, two As, and two Ts.

Total number of arrangements of the letters of the word MATHEMATICS $=\frac{11 !}{2 ! 2 ! 2 !}$

Number of words in which the first word is fixed as $C=$ Number of arrangements of the remaining 10 letters, of which there are two As, two Ms and two Ts $=\frac{10 !}{2 ! 2 ! 2 !}$

Number of words in which the first word is fixed as $T=$ Number of arrangements of the remaining 10 letters, of which there are two As and two $\mathrm{Ms}=\frac{10 !}{2 ! 2 !}$