Question

Find the number of permutations of the word MATHEMEATICS which starts with consonants only.

Answer 1

The word MATHEMATICS has 11 letters.

Of them there are 7 consonants:2 M’s, 2 T’s,H,C,S and 4 vowels:2A’s,E,I

If the word starts with M: As there are 2 M’s it can start wuth M in 2 ways.We will be left with 10 letters out of which A & T come twice. This can be done in 10!/(2!×2!)10!/(2!×2!) ways. So there are 2×10!/(2!×2!)2×10!/(2!×2!) ways

Similarly if it starts with T there will be 2×2×10!/(2!×2!)10!/(2!×2!) ways.

If it starts with H/C/S: In each of the 3 cases we will be left with 10 lettersout of which M,A,T come twice. This can be done in 10!/(2!×2!×2!) ways. So there are 3×10!/(2!×2!×2!) ways.

Total number of permutations:

2×10!/(2!×2!)+2×10!/(2!×2!)+3×10!/(2!×2!×2!)

=10!(1/2+1/2+3/8)

=10!×11/8

=4989600