To solve this problem, we need to distribute 10 cookies into 5 jars (A, B, C, D, E) such that exactly 2 of these jars contain exactly 2 cookies each. Here’s the approach to find the solution:

1. Choose the two jars that will contain exactly 2 cookies.
   We can select 2 jars out of the 5 in $\binom{5}{2}$ ways. So, the number of ways to choose the jars with exactly 2 cookies is: $\binom{5}{2} = 10$

2. Distribute cookies in the chosen jars.
   Since we have 2 jars that must each contain exactly 2 cookies, this accounts for $2 \times 2 = 4$ cookies.

3. Distribute the remaining 6 cookies in the remaining 3 jars.
   The remaining 3 jars can each have any number of cookies, including zero, as long as the sum is 6. We need to count the number of ways to distribute 6 indistinguishable cookies into 3 distinguishable jars, which is a classic "stars and bars" problem.

   The formula for distributing $n$ indistinguishable items into $r$ distinguishable bins is given by: $\binom{n + r - 1}{r - 1}$ Here, $n = 6$ and $r = 3$, so the number of ways to distribute the remaining 6 cookies into 3 jars is: $\binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} = 28$

4. Combine the results.
   Multiply the ways to choose the jars with exactly 2 cookies by the ways to distribute the remaining cookies among the other jars: $10 \times 28 = 280$

Final Answer:

There are 280 ways to distribute the cookies so that exactly 2 jars contain exactly 2 cookies.

Would you like more details or have any questions about this solution?

Here are some related questions for further practice:

1. How many ways can you distribute 10 cookies into 5 jars if there are no restrictions?
2. What if exactly 3 jars must contain exactly 2 cookies each?
3. How would the answer change if no jar can be empty?
4. What if there are 12 cookies and we want exactly 3 jars to contain exactly 3 cookies each?
5. How does the approach change if each jar has a maximum capacity of 4 cookies?

Tip: When distributing indistinguishable items into distinguishable bins, the "stars and bars" method is often a useful tool for counting possibilities.