To determine the number of ways to get a prime sum when rolling three six-sided dice, we need to calculate the possible sums from the dice rolls and count how many of these sums are prime numbers.

Step 1: Possible Sums

Each die can show a number from 1 to 6. When rolling three dice, the minimum possible sum is $1 + 1 + 1 = 3$, and the maximum possible sum is $6 + 6 + 6 = 18$.

Step 2: Prime Numbers Between 3 and 18

The prime numbers between 3 and 18 are:

• 3
• 5
• 7
• 11
• 13
• 17

Step 3: Count Combinations That Yield Prime Sums

To find the number of ways to get each prime sum, we can list all combinations of three dice rolls (1 through 6 on each die) and count the occurrences of the prime sums.

Let's compute the number of ways to get each prime sum:

Prime Sum = 3

• Possible combination: (1, 1, 1) (1 way)

Prime Sum = 5

• Possible combinations: (1, 1, 3), (1, 3, 1), (3, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2, 1) (6 ways)

Prime Sum = 7

• Possible combinations: (1, 1, 5), (1, 5, 1), (5, 1, 1), (1, 2, 4), (1, 4, 2), (2, 1, 4), (2, 4, 1), (4, 1, 2), (4, 2, 1), (1, 3, 3), (3, 1, 3), (3, 3, 1), (2, 2, 3), (2, 3, 2), (3, 2, 2) (15 ways)

Prime Sum = 11

• Possible combinations: (1, 4, 6), (1, 6, 4), (4, 1, 6), (4, 6, 1), (6, 1, 4), (6, 4, 1), (1, 5, 5), (5, 1, 5), (5, 5, 1), (2, 3, 6), (2, 6, 3), (3, 2, 6), (3, 6, 2), (6, 2, 3), (6, 3, 2), (2, 4, 5), (2, 5, 4), (4, 2, 5), (4, 5, 2), (5, 2, 4), (5, 4, 2), (3, 3, 5), (3, 5, 3), (5, 3, 3) (27 ways)

Prime Sum = 13

• Possible combinations: (1, 6, 6), (6, 1, 6), (6, 6, 1), (2, 5, 6), (2, 6, 5), (5, 2, 6), (5, 6, 2), (6, 2, 5), (6, 5, 2), (3, 4, 6), (3, 6, 4), (4, 3, 6), (4, 6, 3), (6, 3, 4), (6, 4, 3), (3, 5, 5), (5, 3, 5), (5, 5, 3), (4, 4, 5), (4, 5, 4), (5, 4, 4) (21 ways)

Prime Sum = 17

• Possible combinations: (6, 6, 5), (6, 5, 6), (5, 6, 6) (3 ways)

Total Number of Ways

Adding all the ways together: $1 + 6 + 15 + 27 + 21 + 3 = 73 \text{ ways}$

So, there are 73 ways to get a prime sum when rolling three six-sided dice.

Would you like a detailed explanation of any step, or have any questions?

Here are some related questions:

1. What is the probability of getting a prime sum when rolling three dice?
2. How would the number of ways change if using dice with different numbers of sides?
3. Can this approach be generalized for rolling more than three dice?
4. What are the most and least probable prime sums with three dice?
5. How does the number of prime sums change with the number of dice?

Tip: When working with probabilities and dice, consider using the multiplication rule to find the total number of outcomes first and then count the specific successful outcomes to simplify your calculations!