5 times 5

5 ! ( 5 1 ) ! ( 5 2 ) ! ( 5 3 ) ! ( 5 4 ) ! = ? \large 5! - (5-1)! - (5-2)! - (5-3)!- (5-4)! = \, ?

Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × × 8 8! = 1\times2\times3\times\cdots\times8 .

24 25 120 5 87

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3 solutions

Hung Woei Neoh
Aug 3, 2016

5 ! ( 5 1 ) ! ( 5 2 ) ! ( 5 3 ) ! ( 5 4 ) ! = 5 ! 4 ! 3 ! 2 ! 1 ! = 5 ( 4 ! ) 4 ! 3 ! 2 ! 1 ! = 4 ( 4 ! ) 3 ! 2 ! 1 ! = 16 ( 3 ! ) 3 ! 2 ! 1 ! = 15 ( 3 ! ) 2 ! 1 ! = 45 ( 2 ! ) 2 ! 1 ! = 44 ( 2 ! ) 1 ! = 88 ( 1 ! ) 1 ! = 87 ( 1 ! ) = 87 5! - (5-1)!-(5-2)!-(5-3)!-(5-4)!\\ =5!-4!-3!-2!-1!\\ =5(4!)-4!-3!-2!-1!\\ =4(4!)-3!-2!-1!\\ =16(3!)-3!-2!-1!\\ =15(3!)-2!-1!\\ =45(2!)-2!-1!\\ =44(2!)-1!\\ =88(1!)-1!\\ =87(1!)\\ =\boxed{87}

It's a permutation problem so: 5! + 4! + 3! + 2! + 1! =120 -24 -6 -2 -1 =87

Typo: 5 ! 4 ! 3 ! 2 ! 1 ! 5! \color{#D61F06}{-}4!\color{#D61F06}{-}3!\color{#D61F06}{-}2!\color{#D61F06}{-}1!

Hung Woei Neoh - 4 years, 10 months ago
Jesse Nieminen
Aug 17, 2016

5 ! ( 5 1 ) ! ( 5 2 ) ! ( 5 3 ) ! ( 5 4 ) ! = 5 ! 4 ! 3 ! 2 ! 1 ! = 120 24 6 2 1 = 87 5! - (5-1)! - (5-2)! - (5-3)!- (5-4)! = 5! - 4! - 3! - 2! - 1! = 120 - 24 - 6 - 2 - 1 = \boxed{87}

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