(i) How many arrangements can be made by using all the letters of the word 'MATHEMATICS'?
(ii) How many of them begin with C?
(iii) How many of them begin with T?

Answer

(i) 4989600 (ii) 453600 (iii) 907200

Explanation

(i) The given word 'MATHEMATICS' has 11 letters, out of which there are 2 M's, 2 A's, 2 T's and the rest are all distinct.
Number of their arrangements `=(11!)/((2!)xx(2!)xx(2!))=4989600.`
(ii) Let each arrangement begin with C. Then, out of the remaining 10 letters, there are 2 M's, 2 A's, 2 T's and the rest are all distinct.
Number of all such arrangements `=(10!)/((2!)xx(2!)xx(2!))=453600.`
(iii) Let each arrangement begin with T. Then, out of the remaining 10 letters, there are 2 M's, 2 A's and the rest are all distinct.
Number of all such arrangements `=(10!)/((2!)xx(2!))=907200.`