How many ways are there to roll doubles or a sum divisible by three on two dice?
To find the ways to roll doubles or a sum divisible by three on two dice, we can consider the following cases:
1) Doubles: There are 6 ways to roll doubles since we have 6 possible outcomes when both dice show the same number (1-1, 2-2, 3-3, 4-4, 5-5, 6-6).
2) Sum divisible by 3: In this case, we need to determine the pairs of numbers that add up to a multiple of 3.
For the first die:
- When the first die shows 1, the second die can show 2 or 5.
- When the first die shows 2, the second die can show 1, 4, or 6.
- When the first die shows 3, the second die can show 1, 4, or 6.
- When the first die shows 4, the second die can show 2 or 5.
- When the first die shows 5, the second die can show 1, 4, or 6.
- When the first die shows 6, the second die can show 2 or 5.
Therefore, there are 2 pairs for case 1, 3 pairs for cases 2-5, and 2 pairs for case 6, giving us a total of 2 + 3(4) + 2 = 16 pairs.
To find the total number of ways to roll doubles or a sum divisible by three, we add the number of ways from both cases:
Total ways = Doubles + Sum divisible by 3
= 6 + 16
= 22
Hence, there are 22 ways to roll doubles or a sum divisible by three on two dice.
To find the number of ways to roll doubles or a sum divisible by three on two dice, we can divide the problem into two parts: finding the number of ways to roll doubles, and finding the number of ways to roll a sum divisible by three.
1. Number of Ways to Roll Doubles:
Since there are 6 possible outcomes on each die, the number of ways to roll doubles is simply 6.
2. Number of Ways to Roll a Sum Divisible by Three:
To count the number of ways to roll a sum divisible by three, we need to analyze the possible combinations of numbers on the two dice.
- If the first die shows 1, the second die can show 2, 5, or 8 to make a sum divisible by three. This gives us 3 possibilities.
- If the first die shows 2, the second die can show 1, 4, 7, 10, or 13. However, 4 and 10 are not possible outcomes since we only have 6-sided dice. Therefore, there are 3 possibilities.
- If the first die shows 3, the possible values on the second die are 3, 6, 9, or 12. This gives us 4 possibilities.
- Similarly, if the first die shows 4, the second die can show 4, 7, or 10 (3 possibilities).
- If the first die shows 5, the second die can show 2, 5, or 8 (3 possibilities).
- If the first die shows 6, the possible values on the second die are 3, 6, 9, or 12 (4 possibilities).
Adding up all the possibilities, we get 3 + 3 + 4 + 3 + 3 + 4 = 20.
Therefore, the total number of ways to roll doubles or a sum divisible by three on two dice is 6 + 20 = 26.
To find the number of ways to roll doubles or a sum divisible by three on two dice, we need to consider each possibility separately.
First, let's find the number of ways to roll doubles. In this case, the value on both dice should be the same.
There are 6 possible outcomes for the first die and 1 outcome for the second die (since it has to match the value of the first die). Therefore, there are 6 possible ways to roll doubles.
Next, let's find the number of ways to get a sum that is divisible by three. To do this, we'll consider the following cases:
Case 1: Both dice show numbers that are multiples of 3.
There are 2 ways to get a multiple of 3 on the first die (3 and 6), and also 2 ways on the second die. So, there are 2 * 2 = 4 ways to get a sum divisible by three in this case.
Case 2: One die shows a value that is a multiple of 3, and the other die shows a value that is not a multiple of 3.
For the first die, there are 2 possibilities (3 and 6). For the second die, there are 4 possibilities (1, 2, 4, and 5) since we cannot use the numbers that are already counted in Case 1. So, there are 2 * 4 = 8 ways to get a sum divisible by three in this case.
Adding up the two cases, we have a total of 4 + 8 = 12 ways to get a sum divisible by three.
Finally, let's compute the total number of ways to roll doubles or a sum divisible by three. To do this, we add up the number of ways for each case.
Therefore, the total number of ways to roll doubles or a sum divisible by three on two dice is 6 + 12 = 18 ways.