Multiple choice

A clinic assigns each patient a code that is either a single vowel (A, E, I, O, U) or a pair of distinct vowels written in alphabetical order. Each patient must receive a different code. If the clinic must be able to assign codes to at least 10 patients, then what is the least number of vowels out of the given five vowels that must be used?

  1. 1

  2. 2

  3. 3

  4. 4

  5. 5
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D Correct answer
Explanation

Number of single vowels = n. Number of pairs = nC2. Total codes = n + nC2 = n + n(n-1)/2 = (2n + n^2 - n) / 2 = (n^2 + n) / 2. We need (n^2 + n) / 2 >= 10, so n^2 + n >= 20. If n=4, (16+4)/2 = 10. So 4 vowels are sufficient.

AI explanation

Using n vowels gives n single codes plus nC2 alphabetical pairs, requiring a total of at least 10. Testing n = 4 yields 4 + 4C2 = 4 + 6 = 10 unique codes. Therefore, 4 is the least number of vowels needed.

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