Explanation:

Six letter words with at least two vowels can have 2, 3, 4 or 5 vowels as no letters can be repeated. There are 21 consonants and 5 vowels.All possible cases:

1. vowels and 4 consonants
2. vowels and 3 consonants
3. vowels and 2 consonants
4. vowels and 1 consonant

∴ Number of ways in which this can be done = ⁵C₂ × ²¹C₄ + ⁵C₃ × ²¹C₃ + ⁵C₄ × ²¹C₂ + ⁵C₅ × ²¹C₁

= 10 × 5985 + 10 × 1330 + 5 × 210 + 1 × 21 = 74221

In each of these cases, chosen 6 letters can arrange themselves in 6! Ways.

∴ Total number of ways in which this can be done = 6! × 74221 = 720 × 74221 = 53439120