A fair 6 sided dice is marked with the numbers 1, 2, 3, 4,5, and 6. The dice is rolled three times and the results are added. How many different sums are possible?

I saw this question on a "Simulated TASC Test." The answer key gives the answer as 16 but I drew a table 6 columns across to represent each number and then 3 rows under the number headings to represent each roll of the die. Then adding the sums, there are a total of 18 combinations not 16.

The first possible sum would be 3 as 1+1+1=3. The largest possible sum would be 18 as 6+6+6=18. If you could the sums possible between and including 3 and 18, you get 16 possible sums!

To find the different sums that are possible when rolling a fair 6-sided dice three times, we can analyze the combinations that can be obtained.

First, let's consider the possible results of rolling a single time. The dice has six sides, so there are six possible outcomes: 1, 2, 3, 4, 5, and 6.

When rolling the dice three times, the sum of the results can vary. To calculate the number of different sums, we need to find out how many combinations of numbers can be obtained.

Using all possible combinations, we can determine the minimum and maximum sums:

- The minimum sum would be obtained by rolling three 1s (1 + 1 + 1 = 3).
- The maximum sum would be obtained by rolling three 6s (6 + 6 + 6 = 18).

So, the range of possible sums is from 3 to 18 (inclusive).

To find the number of different sums, we need to consider all possible combinations within this range. We can use combinatorial mathematics to calculate this:

- There are three dice rolls, and each roll has six possible outcomes. Hence, the total number of combinations is 6^3 = 216.

However, some combinations may result in the same sum. For example, 1 + 2 + 3 is equivalent to 3 + 2 + 1. So, we need to consider these cases.

Let's break it down further:

- There is only one way to obtain a sum of 3 (rolling three 1s).
- There are three ways to obtain a sum of 4 (1 + 1 + 2, 1 + 2 + 1, 2 + 1 + 1).
- Similarly, there will be three ways to obtain a sum of 18 (rolling three 6s).

As we move towards the middle sums, such as 9, we can see that there are multiple combinations possible:

- One way to approach this is to use an observation. We notice that the minimum sum (3) and the maximum sum (18) are unique. However, the sums that lie between them will have multiple combinations.

- Another approach is to use the concept of combinations. For a single roll, there are six possible outcomes. Since there are three rolls, there would be 6^3 = 216 total combinations. From this, we can deduct the unique sums (3 and 18). We also subtract the possible sums with repeated numbers. For example, for 9, there is only one combination where three numbers are the same (3 + 3 + 3), so we count it as one combination rather than three. This pattern holds true for other sums.

Hence, the number of different sums that can be obtained by rolling a fair 6-sided dice three times is:

1 (sum of 3) + 3 (sum of 4) + 5 (sum of 5) + 7 (sum of 6) + 9 (sum of 7) + 11 (sum of 8) + 11 (sum of 9) + 9 (sum of 10) + 7 (sum of 11) + 5 (sum of 12) + 3 (sum of 13) + 1 (sum of 14) = 76

Therefore, there are 76 different possible sums.