Answer 1

In the word 'ALGORITHM' the number of letters is n = 9.

No two vowels are together:

Consider one of the possible arrangements of V (vowels) and C (consonants)

VCVCVCVCVCVCV

Here, there are 7 positions for vowels and 6 positions for consonants.

∴ 3 vowels can be arranged in ⁷P₃ ways and 6 consonants can be arranged in ⁶P₆ ways.

Hence, the total number of words in which no two vowels are together

= ⁷P₃ × ⁶P₆

= `(7!)/(4!) xx 6!`

= 7 × 6 × 5 × (6 × 5 × 4 × 3 × 2 × 1)

= 210 × 720

= 151200