# Find $a,b,c$ such that $a!\times b!\times c!=d!$ [duplicate]

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I have to find $a,b,c \in \mathbb{N}$ such that-$a!\times b!\times c!=d!$

Answer given in my book is $3!\times5!\times7!=10!$(But it is written that other answers are also possible).

What is a systematic way to find the answer without using a calculator or computer?(I mean without brute force calculation)

Thanks for any help!!

## marked as duplicate by Jyrki LahtonenMar 13 '16 at 7:59

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• One set of solutions is a,b,c and d are in the set {0,1}. – user145600 Mar 13 '16 at 7:36
• – mathlove Mar 13 '16 at 7:37
• There are many uninteresting answers, such as $12!=11!\cdot 12=11!\cdot 2!\cdot 3!$. – André Nicolas Mar 13 '16 at 7:41
• @AndréNicolas Your answer provides one way of approaching the solution....But you are fortunate that 12 can be expressed as a product of two factorials...not all numbers can be expressed as a product of two factorials:-)... – tatan Mar 13 '16 at 7:44
• @tatan The point is we can always take $a=b!c!-1,\,d=b!c!$ for a trivial solution. – J.G. Mar 13 '16 at 7:53