The 11 number of arrangements that can be made with the letters of the word 'MATHEMATICS' in which all vowels come together, is

There are 4 vowels viz. A, E, A, I. Considering these four vowels as one letter we have 8 letters (M, T, H, M, T .. C, S and one letter obtained by combining all vowels), out or which M occurs twice, T occurs twice and the rest all different. These 8 letters can be arranged in 8!2!×2! ways.But, the four vowels (A, E, A, I) can be put together in 4!2! ways.Hence, the total number of arrangements in which vowels are alwavs together =8!2!×2!×4!2!.