Mathematics - Permutation & Combination Question with Solution | TestHub

The letters other than vowels are: PRMTTN Number of permutations with no two vowels together is $\frac{6!}{2!}\times {}^7{\mathrm{C}}_5\times 5!$ Further among these permutations the number of cases in which T's are together is $5!\times {}^6{\mathrm{C}}_5\times 5!$ So the required number $=\frac{6!}{2!}{}^7{\mathrm{C}}_5\times 5!-5!\times {}^6{\mathrm{C}}_5\times 5!=57\times (5!{)}^2$