How many strings of English letters of length 8 contain at least 6 vowels

combinatorics

I am having a hard time coming up with a way to calculate this.

There are 5 vowels so I think the number of strings that contain exactly 1 vowel would be:

$5$ × $8$ × $21^5$

But I am not sure where to go from here. I believe I need to find a number to subtract from this number.

Best Answer

There are

  • $\binom825^621^2$ admissible strings with $6$ vowels ($\binom82$ ways to locate the consonants, then $5^6$ choices for the vowels and $21^2$ choices for the consonants)
  • $\binom815^721^1$ admissible strings with $7$ vowels
  • $\binom805^821^0$ admissible strings with $8$ vowels

Thus there are $206453125$ admissible strings in total.

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