What is the minimum number of distinct letters needed to create unique identification codes for 60 participants, using combinations of one to three letters?

To determine the minimum number of distinct letters needed to create unique identification codes for 60 participants using combinations of one to three letters, we can calculate the total number of combinations that can be formed with different counts of letters.

1. Combinations of One Letter: With ( n ) letters, the number of unique combinations of one letter is simply ( n ).

2. Combinations of Two Letters: The number of combinations of two letters can be calculated as ( n^2 ), as each letter can be followed by any of the ( n ) letters.

3. Combinations of Three Letters: The combinations of three letters is calculated as ( n^3 ), where each of the three positions can be filled by any of the ( n ) letters.

Hence, the total number of unique identification codes that can be generated using ( n ) letters is:

[

\text{Total Codes} = n + n^2 + n^3

]

Next, we set up the inequality to find the minimum number of letters required:

[

n + n^2 + n^3 \geq 60

]

Now, let’s evaluate this expression for different values of