Compute \(\large \dfrac{(10!\color{green}{+}9!)(8!\color{green}{+}7!)(6!\color{green}{+}5!)(4!\color{green}{+}3!)(2!\color{green}{+}1!)}{(10!\color{red}{-}9!)(8!\color{red}{-}7!)(6!\color{red}{-}5!)(4!\color{red}{-}3!)(2!\color{red}{-}1!)} = \, \, \large ?\)
+23245 Numerator: 10! + 9! = 10·9! + 9! = 10·9! + 1·9! = (10 + 1)·9! = 11·9!
8! + 7! = 8·7! + 7! = 8·7! + 1·7! = (8 + 1)·7! = 9·7!
6! + 5! = 6·5! + 5! = 6·5! + 1·5! = (6 + 1)·5! = 7·5!
4! + 3! = 4·3! + 3! = 4·3! + 1·3! = (4 + 1)·3! = 5·3!
2! + 1! = 2·1! + 1! = 2·1! + 1·1! = (2 + 1)·1! = 3·1!
So, the numerator becomes: 11·9! · 9·7! · 7·5! · 5·3! · 3·1!
Denominator: 10! - 9! = 10·9! - 9! = 10·9! - 1·9! = (10 - 1)·9! = 9·9!
Similarly: 8! - 7! = 7·7!
6! - 5! = 5·5!
4! - 3! = 3·3!
2! - 1! = 1·1!
The denominator becomes: 9·9! · 7·7! · 5·5! · 3·3! · 1·1!
Start cancelling out, and you'll get the (rather surprising) short answer.
