To many exclamations!

Number Theory Level 3

60 ! ! ! 60 ! ! ! ! ! ! = ( a − b ) ! ! ! ! ! ! \large\dfrac{60!!!}{60!!!!!!}=(a-b)!!!!!!

Evaluate ( a − b ) (a-b) if the above equation holds true.

Note: ! ! mark denotes factorial.


The answer is 57.

Sravanth C. by Sravanth C. , 21, India

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1 solution

Sravanth C.
Oct 10, 2015

60 ! ! ! 60 ! ! ! ! ! ! = 60 × ( 60 − 3 ) × ( 60 − 6 ) × ( 60 − 9 ) × . . . × 3 60 × ( 60 − 6 ) × ( 60 − 12 ) × ( 60 − 18 ) × . . . × 6 = ( 60 − 3 ) × ( 60 − 9 ) × ( 60 − 15 ) × . . . × 3 = ( 60 − 3 ) ! ! ! ! ! ! \dfrac{60!!!}{60!!!!!!}=\dfrac{60×(60-3)×(60-6)×(60-9)× . . . ×3}{60×(60-6)×(60-12)×(60-18)× . . . ×6}\\=(60-3)×(60-9)×(60-15)× . . . ×3\\\boxed{=(60-3)!!!!!!}

Therefore ( a − b ) = ( 60 − 3 ) = 57 (a-b)=(60-3)=\boxed{57} .

Note: it can also be generalized to the n t h n^{th} term, i.e. n ! ( x ) n ! ( 2 x ) = ( n − x ) ! ( 2 x ) \dfrac{n!^{(x)}}{n!^{(2x)}}=(n-x)!^{(2x)}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

How do you define '!' mark, It is international acceptance that it is factorial, Your question didn't provide its definition .This question is senseless and so is your solution

A Former Brilliant Member - 5 years, 8 months ago

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Done, I've edited it sir. In future if you have any problems, report it in the more menu.

Sravanth C. - 5 years, 8 months ago

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