100 Factorial Factorial

Algebra Level 1

( 100 ! + 99 ! ) × ( 98 ! + 97 ! ) × ( 2 ! + 1 ! ) ( 100 ! 99 ! ) × ( 98 ! 97 ! ) × ( 2 ! 1 ! ) \dfrac{\big(100! + 99!\big) \times \big(98! + 97!\big) \times \ldots \big(2! + 1!\big)}{\big(100! - 99!\big) \times \big(98! - 97!\big) \times \ldots \big(2! - 1!\big)}


Inspiration

101 101 99 99 100 100

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1 solution

Mahdi Raza
Jun 9, 2020

= ( 100 ! + 99 ! ) × ( 98 ! + 97 ! ) × ( 2 ! + 1 ! ) ( 100 ! 99 ! ) × ( 98 ! 97 ! ) × ( 2 ! 1 ! ) = ( 100 × 99 ! + 99 ! ) × ( 98 × 97 ! + 97 ! ) × ( 2 × 1 ! + 1 ! ) ( 100 × 99 ! 99 ! ) × ( 98 × 97 ! 97 ! ) × ( 2 × 1 ! 1 ! ) = 99 ! ( 100 + 1 ) 99 ! ( 100 1 ) 97 ! ( 98 + 1 ) 97 ! ( 98 1 ) 1 ! ( 2 + 1 ) 97 ! ( 2 1 ) = 101 99 99 97 3 1 = 101 \begin{aligned} &= \dfrac{\big(100! + 99!\big) \times \big(98! + 97!\big) \times \ldots \big(2! + 1!\big)}{\big(100! - 99!\big) \times \big(98! - 97!\big) \times \ldots \big(2! - 1!\big)} \\ \\ &= \dfrac{\big(100 \times 99! + 99!\big) \times \big(98 \times 97! + 97!\big) \times \ldots \big(2 \times 1! + 1!\big)}{\big(100 \times 99! - 99!\big) \times \big(98 \times 97! - 97!\big) \times \ldots \big(2 \times 1! - 1!\big)} \\ \\ &= \dfrac{\cancel{99!} (100+1)}{\cancel{99!} (100-1)} \cdot \dfrac{\cancel{97!} (98+1)}{\cancel{97!} (98-1)} \cdots \dfrac{\cancel{1!} (2+1)}{\cancel{97!} (2-1)} \\ \\ &= \dfrac{101}{\cancel{99}} \cdot \dfrac{\cancel{99}}{\cancel{97}} \cdots \dfrac{\cancel{3}}{1} \\ \\ &= \boxed{101} \end{aligned}

Can you elaborate on the second to last line?

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It forms a telescopic product, so all the terms cancel out

Mahdi Raza - 1 year ago

I have to say this- This is a very elegant problem, so much fun to solve when so many terms cancel out!

Vinayak Srivastava - 1 year ago

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Yes it is, BTW this is not my original idea, it is just a variation from the inspiration because someone could have brute-forced that problem

Mahdi Raza - 1 year ago

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I actually first solved the inspiration with brute force only, but then realized how weird my thinking was and attempted yours by myself!

Vinayak Srivastava - 1 year ago

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@Vinayak Srivastava Haha, I hope you learned a new trick!

Mahdi Raza - 1 year ago

Awesome problem! Awesome solution! Awesome Everything! I love it when something in math looks really complex until you simplify it. This is the beauty of math. Awesome!

A Former Brilliant Member - 11 months, 3 weeks ago

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Thank you Yashvardhan. Maths is very beautiful indeed

Mahdi Raza - 11 months, 3 weeks ago

By the way, what does using brute-force to solve a problem mean?? You guys use it a lot.

A Former Brilliant Member - 11 months, 3 weeks ago

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Checking it out every option / case there could be

Mahdi Raza - 11 months, 3 weeks ago

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Ok, thanks.

A Former Brilliant Member - 11 months, 3 weeks ago

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