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For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This problem is an application of the non negative integer solution and the symmetry argument. This is from ISI MStat 2015 PSB Problem 4.

Suppose 15 identical balls are placed in 3 boxes labeled A, B and C. What is the number of ways in which Box A can have more balls than

Box C?

- Symmetry Argument
- Number of non negative integer solutions to \(x+y+z = 15\) is \(17 \choose 2\). Not sure why? Search it up!

There are three possible cases.

- Box A has more balls than Box C.
- Box C has more balls than Box A.
- Box A has the same number of balls as Box C.

**The symmetry argument**

The number of ways in which Box A has more balls than Box C = The number of ways in which Box C has more balls than Box A.

Isn't that very obvious, since the balls are not biased towards boxes. Why will they be?

Total Number of Ways = The number of ways in which Box A has more balls than Box C + The number of ways in which Box C has more balls than Box A + The number of ways in which Box A has the same number of balls as Box C.

The number of ways in which Box A has the same number of balls as Box C = 8 right?

Why? They can have either 0 ball each, 1 ball each, 2 balls each, ..., 7 balls each at most.

Therefore, The number of ways in which Box A has more balls than Box C = \(\frac{{17 \choose 2 }- 8}{2} = 64 = \frac{(n+1)^2}{2}\). [\( n = 15\)]

Suppose 15 identical balls are placed in 3 boxes labeled A, B, and C.

What is the number of ways in which Box A have no fewer balls than

Box B and Box B have no fewer balls than Box C?

This is related to the topic of mathematics called **Partitions of Numbers**.

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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