How many arrangements can be made with the letters of the word MATHEMATICS ? In how many twice ?

Explanation

There are 11 letters in the word MATHEMATICS. M occurs twice, T occurs twice, A occurs twice and the rest all are different
`therefore` Required number of arrangements `=(11!)/(2!2!2!)=4989600`
The consonants are M occurs twice, T occurs twice and H,C,S occur one time. Considering these seven consoants as one letter. Therefore, we have to find the arrangement of 4 vowels (A,E,A,I) and a group of consonants.
`therefore ` Number of arragements `=(5!)/(2!)=60`
The seven consonants can also be arranged itself in `(7!)/(2!2!)=1260` ways
`therefore` Required number of arrangements =`1260xx60=75600`