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The problem statement is as follows: "Harry and Ron were given 8 different pens and 6 different pencils by their mother. In how many ways can the pens and pencils be distributed among them such that each of them gets at least three pens and two pencils?"

To solve this, we need to consider the distribution of both pens and pencils:

1. **Distributing the Pens**:
   - Each must receive at least 3 pens. We distribute 3 pens to Harry and 3 to Ron. There are 2 pens left to distribute.
   - We can use the "stars and bars" theorem for distributing these 2 pens. Harry and Ron could receive \(0, 1,\) or \(2\) additional pens, but both can't receive \(0\). The possibilities are: \( (2, 0), (1, 1), (0, 2) \).
   - For each scenario, the number of ways to choose the pens for Harry and Ron from the remaining two pens is \(\binom{2}{2}, \binom{2}{1},\) and \(\binom{2}{0}\) respectively.
   - Additionally, the first 3 pens can be any 3 of the 8 pens, and the same for Ron. So we calculate: \(\binom{8}{3} \times \binom{5}{3}\) to distribute the initial 3 pens each, and then multiply by the distribution of the last 2 pens.

2. **Distributing the Pencils**:
   - Similar to the pens, each must receive at least 2 pencils. We distribute 2 pencils to Harry and 2 to Ron. There is 1 pencil left to distribute.
   - The remaining pencil can either go to Harry or Ron, i.e., 2 possibilities.
   - For the initial 2 pencils each, the calculation is \(\binom{6}{2} \times \binom{4}{2}\).

Combining the two distributions (multiplying the ways for pens and pencils), we calculate the total number of ways Harry and Ron can receive the pens and pencils. Let's compute the exact number.The total number of ways in which Harry and Ron can be given the pens and pencils such that each of them gets at least three pens and two pencils is 403,200.

Would you like more details or do you have any questions about this problem? Here are some related questions:

1. What if the number of pens or pencils changes, how would that affect the distribution?
2. Can you explain the "stars and bars" theorem used in this distribution?
3. What if there was a minimum but no maximum number of pens or pencils that could be distributed?
4. How does the calculation change if the pens or pencils are not distinct?
5. What other problems can be solved using similar combinatorial methods?

**Tip**: When solving combinatorial distribution problems, it's useful to ensure all constraints are considered (like minimum distributions), and apply principles like the "stars and bars" theorem to find the number of ways items can be distributed.

Harry and Ron were given 8 different pens and 6 different pencils by their mother. In how many ways can the pens and pencils be distributed among them such that each of them gets at least three pens and two pencils?

This combinatorics problem involves distributing 8 pens and 6 pencils between two people, with restrictions that each person must receive at least three pens and two pencils. Learn how to apply the stars and bars theorem and combination formulas to solve this complex distribution problem.

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