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Divisibility Practice

Prove that \((2014!)!\) is divisible by \( ((2014 \cdot 2013)!)^{2012!}\)

Note by Pratik Shastri
2 years, 7 months ago

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Let us have \(2014!\) balls. Let us divide them in \(2012!\) groups, each group having \(2013.2014\) balls.

Let's assume that balls of each group are of different colour, but all balls of the same group are the of the same colour.

Number of ways of doing so is, \[\dfrac{(2014!)!}{((2014.2013)!)^{2012!}}\] Number of ways of doing something is always a positive integer. Hence Proved. \(\boxed{.}\) Satvik Golechha · 2 years, 7 months ago

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@Satvik Golechha Using Combinatorics for a Number Theory Problem is simply amazing! Marc Vince Casimiro · 2 years, 7 months ago

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@Satvik Golechha Excellent! +1 Pratik Shastri · 2 years, 7 months ago

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@Satvik Golechha @Satvik Golechha this is great method see solution of @John Ashley Capellan here Shivamani Patil · 2 years, 7 months ago

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@Satvik Golechha superb way to do this...... Rajat Bhagat · 2 years, 6 months ago

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