To solve this problem, we need to distribute 10 cookies among 5 jars (A, B, C, D, E) with the constraint that exactly two of the jars contain exactly 2 cookies. The remaining cookies can be distributed among the other jars, with some possibly holding zero cookies. Here’s a step-by-step solution:

Step 1: Choosing Jars with Exactly 2 Cookies

Since exactly two jars must contain exactly 2 cookies, we first select the two jars that will hold these 2 cookies each. The number of ways to choose 2 jars from 5 is given by the combination: $\binom{5}{2} = 10$

Step 2: Distributing Remaining Cookies

After placing 2 cookies in each of two jars, we have used up $2 \times 2 = 4$ cookies. This leaves $10 - 4 = 6$ cookies to be distributed among the remaining three jars, with no restrictions on how many each jar can contain (including zero).

The problem now reduces to finding the number of ways to distribute 6 indistinguishable cookies among 3 jars. This is a "stars and bars" problem, which is solved using the formula for distributing $n$ indistinguishable items into $k$ distinguishable bins: $\binom{n + k - 1}{k - 1} = \binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} = 28$

Step 3: Calculating Total Arrangements

Since there are 10 ways to choose the jars with exactly 2 cookies, and for each choice, there are 28 ways to distribute the remaining cookies, the total number of ways is: $10 \times 28 = 280$

Answer

The number of ways to distribute the cookies so that exactly two jars contain exactly 2 cookies is 280.

Would you like further details on any part of this solution?

Here are some related questions to explore:

1. What if three jars were required to contain exactly 2 cookies each?
2. How does the solution change if all jars must contain at least one cookie?
3. How would we solve it if we had distinguishable cookies?
4. What if we increased the total number of cookies to 15 but kept the same jar setup?
5. How does this problem relate to the concept of "integer partitions"?

Tip: In combinatorial problems, separating choices step-by-step helps manage complex constraints and find the solution incrementally.