This is a combinatorics problem that involves distributing 10 apples and 10 pears among three bowls (red, green, and blue), with the restriction that each bowl must contain at least one apple and one pear.

Step 1: Adjust the number of apples and pears

Since each bowl must have at least one apple, we begin by placing one apple in each of the three bowls. This leaves us with:

$10 - 3 = 7 \text{ apples}$

Similarly, since each bowl must have at least one pear, we place one pear in each bowl, leaving us with:

$10 - 3 = 7 \text{ pears}$

Now, we need to find how many ways we can distribute these remaining 7 apples and 7 pears into the three bowls.

Step 2: Use stars and bars

The problem now is to determine how many ways we can distribute the 7 remaining apples into 3 bowls, and separately how many ways we can distribute the 7 remaining pears into the same 3 bowls.

For distributing $n$ indistinguishable objects into $k$ distinguishable bins (bowls), the number of ways is given by the "stars and bars" formula:

$\text{Number of ways} = \binom{n+k-1}{k-1}$

Apples:

For distributing 7 apples into 3 bowls, the number of ways is:

$\binom{7+3-1}{3-1} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$

Pears:

Similarly, for distributing 7 pears into 3 bowls, the number of ways is:

$\binom{9}{2} = 36$

Step 3: Multiply the results

Since the distributions of apples and pears are independent of each other, the total number of ways to distribute the apples and pears is the product of the two results:

$36 \times 36 = 1296$

Final Answer:

The total number of ways Nick can distribute 10 apples and 10 pears into the three bowls, such that each bowl contains at least one apple and one pear, is $\boxed{1296}$.

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Would you like further details on any part of the solution? Here are 5 related questions you can explore:

1. How does the stars and bars method work for different numbers of bowls?
2. What if there were 12 apples and 12 pears? How would the calculation change?
3. What if the apples and pears were distinguishable (e.g., labeled)?
4. Can this problem be solved using generating functions?
5. How does the result change if one bowl is allowed to be empty?

Tip: When dealing with combinatorics problems that have restrictions, a good strategy is to handle the restrictions first before applying standard counting techniques.