Question

A computer randomly prints three-digit codes, with no repeated digits in any code. What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical

Answer 1

Using the pigeonhole principle, since there are 720 unique three-digit codes without repeated digits, to guarantee at least six identical codes, we must print all unique codes plus 5 more, totaling 725 codes.

Step-by-step explanation:

To determine the minimum number of three-digit codes a computer must print to guarantee at least six codes are identical, we need to consider the worst-case scenario using the principle of the pigeonhole principle. The pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. Since the codes do not have repeated digits, each digit can only be used once per code.

There are 10 possible digits (0-9) for each place in the code, but since no digit can repeat in a single code, the first digit can be any of the 10, the second can only be one of the remaining 9, and the third can only be one of the remaining 8. This leads to a total of 10 x 9 x 8 = 720 unique codes without repetition of digits.

To ensure at least six identical codes, we must exhaust all unique possibilities plus one additional set of five codes to have the sixth identical code. Therefore, the minimum number of codes printed would have to be 720 unique codes + 5 additional (identical) codes, resulting in 725 minimum codes to guarantee that at least six are identical.

Answer 2

To guarantee at least six identical three-digit codes without repeated digits, a minimum of 3241 codes must be printed, deriving from the calculation of 9 × 9 × 8 times 5 plus 1.

Step-by-step explanation:

The question aims to find the minimum number of three-digit codes that must be randomly generated by a computer to guarantee that at least six codes are identical, assuming no digit repeats within a code. Since there are 10 possible digits (0-9), but the first digit cannot be 0 (to make a three-digit number), there are 9 options for the first digit, 9 remaining options for the second digit (since the first digit cannot repeat), and 8 options for the third digit (as neither the first nor second digit can repeat). Therefore, there are 9 × 9 × 8 unique three-digit codes possible, which equals 648.

By the Pigeonhole Principle, to ensure that at least one code has been printed six times, we must exceed the total number of possible unique codes times five (since five of the same code would not be enough - the sixth guarantees a repeat). That means the minimum number that must be printed is 648 × 5 plus 1, which equals 3241.