Sarah and Sarat have 12 marbles together. List all the ways they can share these marbles.

Sarah and Sarat have a total of 12 marbles. We need to find all possible ways they could share these marbles, meaning we need to find pairs of numbers that add up to 12.

Let $S$ be the number of marbles Sarah has and $R$ be the number of marbles Sarat has.
We know that $S + R = 12$. We will list all non-negative integer pairs $(S, R)$ that satisfy this equation.

• If Sarah has 0 marbles, Sarat has $12 - 0 = 12$ marbles. (0, 12)
• If Sarah has 1 marble, Sarat has $12 - 1 = 11$ marbles. (1, 11)
• If Sarah has 2 marbles, Sarat has $12 - 2 = 10$ marbles. (2, 10)
• If Sarah has 3 marbles, Sarat has $12 - 3 = 9$ marbles. (3, 9)
• If Sarah has 4 marbles, Sarat has $12 - 4 = 8$ marbles. (4, 8)
• If Sarah has 5 marbles, Sarat has $12 - 5 = 7$ marbles. (5, 7)
• If Sarah has 6 marbles, Sarat has $12 - 6 = 6$ marbles. (6, 6)
• If Sarah has 7 marbles, Sarat has $12 - 7 = 5$ marbles. (7, 5)
• If Sarah has 8 marbles, Sarat has $12 - 8 = 4$ marbles. (8, 4)
• If Sarah has 9 marbles, Sarat has $12 - 9 = 3$ marbles. (9, 3)
• If Sarah has 10 marbles, Sarat has $12 - 10 = 2$ marbles. (10, 2)
• If Sarah has 11 marbles, Sarat has $12 - 11 = 1$ marble. (11, 1)
• If Sarah has 12 marbles, Sarat has $12 - 12 = 0$ marbles. (12, 0)

The possible numbers of marbles they could have each are:

The possible ways they could have marbles are:
Sarah could have 0 and Sarat 12.
Sarah could have 1 and Sarat 11.
Sarah could have 2 and Sarat 10.
Sarah could have 3 and Sarat 9.
Sarah could have 4 and Sarat 8.
Sarah could have 5 and Sarat 7.
Sarah could have 6 and Sarat 6.
Sarah could have 7 and Sarat 5.
Sarah could have 8 and Sarat 4.
Sarah could have 9 and Sarat 3.
Sarah could have 10 and Sarat 2.
Sarah could have 11 and Sarat 1.
Sarah could have 12 and Sarat 0.

These are all the possible distributions.