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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?

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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.

10 Cookies have to be kept in 5 jars A, B, C, D, E. A jar may contain any number of cookies (including 0). What is the number of ways to distribute so that exactly 2 jars contain exactly 2 cookies.

Explore the solution to distributing 10 cookies across 5 jars with the requirement that exactly 2 jars contain exactly 2 cookies each. This combinatorial problem involves combinations and the stars and bars method, suitable for students in Grades 11-12.

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