Question

Hobart bus tickets have numbers from 000000 to 999999. The number on the first ticket for each

bus is random, with subsequent ticket numbers going up by one each time. In the rare event of
going past 999999, the next number is 000000. A lucky ticket is one where the sum of the digits is
21.
What is the average sum of the digits on a ticket?

Answer 1

To find the average sum of the digits on a ticket, we consider the possible sums of the digits and their frequencies. Using combinatorics, we calculate the frequency of each sum for the six places on the ticket.

Step-by-step explanation:

To find the average sum of the digits on a ticket, we need to consider the possible sums of the digits and their frequencies. Since the sum of the digits can range from 3 to 27, we will calculate the frequency of each possible sum.

There are 10 possible digits (0 to 9), and each digit can appear in each of the six places on the ticket (hundred thousands, ten thousands, thousands, hundreds, tens, units).

Therefore, there are 6^6 = 46656 possible ticket numbers.

To determine the frequency of each sum, we can use combinatorics. For example, for a sum of 21, we choose 3 digits out of the 6 places on the ticket to have a sum of 21, and the remaining 3 digits can have any value.

The number of ways to choose 3 digits out of 6 is given by the combination formula C(6, 3) = 20.

The remaining 3 digits can take 10^3 = 1000 different values. Therefore, the frequency of the sum 21 is 20 * 1000 = 20000.

We can repeat this process for all possible sums.

After calculating the frequencies for all sums from 3 to 27, we can find the average sum by adding up the products of each sum and its frequency, and dividing by the total number of ticket numbers (46656).