To determine how many outcomes have a sum between 3 and 12 when rolling two dice, let's analyze the situation:

Each die can show a number from 1 to 6. The table given shows all 36 possible outcomes (since $6 \times 6 = 36$). We need to count the number of pairs where the sum is between 3 and 12 inclusive. The minimum possible sum is 2 (when both dice show 1) and the maximum is 12 (when both dice show 6).

Step-by-Step Solution:

1. Identify the Range: The problem asks for sums between 3 and 12. Note that every possible sum from rolling two dice is already within this range.
2. Count Valid Outcomes: Since every possible outcome from the dice (ranging from $1+1=2$ to $6+6=12$) can produce a sum of 3 or more, we need to exclude only the outcomes where the sum is 2:
   • The sum of 2 can only occur from the pair (1,1).
3. Calculate: There are 36 total outcomes. Subtract the 1 outcome where the sum is 2.

$\text{Total valid outcomes} = 36 - 1 = 35$

Therefore, there are 35 different ways the sum can be between 3 and 12.

Would you like a more detailed explanation or have any questions?

Here are five related questions to deepen understanding:

1. What is the probability of rolling a sum of 7 with two dice?
2. How many outcomes result in an even sum when rolling two dice?
3. What is the most common sum when rolling two dice, and why?
4. If one die is fixed at 3, how many sums between 3 and 12 are possible?
5. How would the total outcomes change if we used three dice instead of two?

Tip: When calculating probabilities with dice, always start by listing all possible outcomes; it helps in understanding patterns and probability distributions.