Multiple choice

An organizer plans to identify each volunteer at a health-awareness fair with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the minimum number of letters needed if there are 18 volunteers, and each volunteer must receive a different code?

  1. 5

  2. 6

  3. 7

  4. 8

  5. 9
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Codes are single letters (n) or pairs (nC2). Total codes = n + n(n-1)/2 = (2n + n^2 - n)/2 = (n^2 + n)/2. We need (n^2 + n)/2 >= 18. n^2 + n >= 36. If n=5, 25+5=30 < 36. If n=6, 36+6=42 > 36. So 6 letters are needed.

AI explanation

If there are n letters available, the number of single-letter codes is n, and the number of two-letter codes written in alphabetical order is nC2, calculated as n times (n minus 1) divided by 2. Testing n equals 5 gives 5 plus 10, which is 15 codes, falling short of the 18 needed. Testing n equals 6 gives 6 plus 6C2, which is 6 plus 15, resulting in exactly 21 available codes, so the minimum number of letters needed is 6.

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