Multiple choice

A researcher plans to identify each participant in a $174$ certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are $12$ participants, and each participant is to receive a different code?

  1. $4$
  2. $5$
  3. $6$
  4. $7$
  5. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

With n letters, the number of available codes is n + n(n - 1)/2. Four letters give only 10 codes, while five letters give 15, so the least number needed is 5.

AI explanation

If n letters are used, the number of available codes equals the n single letters plus the number of pairs of distinct letters, which is nC2. We need n + nC2 to be at least 12. Testing n = 4 yields 4 + 4C2 = 4 + 6 = 10 codes, which is not enough. Testing n = 5 yields 5 + 5C2 = 5 + 10 = 15 codes, which successfully provides at least 12 different codes.

Keep practicing — related questions