Mangoes-15
Students-3

The no. of ways of mangoes that can be equally shared among three students is 15P3

Understanding the Problem
To divide 15 mangoes equally among three students, we need to determine how many mangoes each student receives and how to arrange them.
Equal Distribution
- Since there are 15 mangoes and 3 students, each student will receive:
- 15 mangoes / 3 students = 5 mangoes per student.
Distribution Combinations
- The problem now is to find out in how many ways we can distribute these mangoes.
- If all mangoes are identical, the distribution is straightforward:
- Each student receives 5 identical mangoes.
Using Stars and Bars Method
- The "stars and bars" theorem can be applied for distribution:
- Here, we treat the mangoes as "stars" and the dividers (between students) as "bars."
- In this case, we consider dividing 15 identical mangoes into 3 groups:
- We need 2 bars to create 3 partitions.
Calculating the Ways
- The total number of objects (stars + bars) is:
- 15 (mangoes) + 2 (bars) = 17.
- The number of ways to arrange these objects is given by the formula:
- C(17, 2) = 17! / (2!(17-2)!) = 136.
Conclusion
- Therefore, the number of ways to equally distribute 15 mangoes among three students is:
- 136 ways, considering all mangoes are identical.
This approach simplifies the distribution process while ensuring that each student receives an equal amount, leading to an even and fair sharing of resources.