Five Zeroes And ! Use To Make 120

USE FIVE ZEROES AND TRIGNO AND MATHS NOTATIONS TO MAKE 120
I really love this puzzle. This is nothing but just requires your presence of mind and some tricky use of mathematical and trigonometric operators.

1.) Make 120 with 5 zeros :-

TRY AND SOLUTIONS SET DEEP BELOW


























Sol 1 :- (0! + 0! + 0! + 0! + 0!)! = 120

Here '!' symbol is used for factorial.
And 0! = 1
= (0! + 0! + 0! + 0! + 0!)!
= (1 + 1 + 1 + 1 + 1)!
= 5!
= 120

Sol 2 :- (cos (0) + cos (0) + cos (0) + cos (0) + cos (0))! = 120

= (cos (0) + cos (0) + cos (0) + cos (0) + cos (0))!
= (1 + 1 + 1 + 1 + 1)!
= 5!
= 120

Similarly you can replace cos (0) with cot (0).

Sol 3 :-(0!0! * 0!0!) - 0! = 120

= (0!0! * 0!0!) - 0!
= (11 * 11) - 1
= 121 - 1
= 120

Sol 4 :- ((0! + 0!)^(0! + 0!) + 0!)! = 120

= ((0! + 0!)^(0! + 0!) + 0!)!
= ((1 + 1)^(1 + 1) + 1)!
= (2^2 + 1)!
= (4 + 1)!
= 5!
= 120

Similarly you can do it in many ways. Seems quite easy. Isn't it. Lets make it a bit difficult.

2.) Make 120 with 4 zeros :-

((0! + 0! + 0!)! - 0!)! = 120

= ((0! + 0! + 0!)! - 0!)!
= ((1 + 1 + 1)! - 1)!
= (3! - 1)!
= (6 - 1)!
= 5!
= 120

3.) Make 120 with 3 zeros :-

Γ((0! + 0! + 0!)!) = 120

Here 'Γ' is called "Gamma Function".
Γ (n) = (n - 1) ! , where n = 0, 1, 2, 3 , .......

= Γ((0! + 0! + 0!)!)
= Γ((1 + 1 + 1)!)
= Γ(3!)
= Γ(6)
= 5!
= 120

4.) Make 120 with 1 zero :-

Sol 1 :- sec ( tan-1 (…… sec ( tan-1 (0!)) ……)), with sec ( tan-1 ….) taken 14399 times = sqrt (14400) = 120

Consider a right-angled triangle with sides (1,1,√2). Thus,
√2 = sec ( tan-1( 0! ) )
Now, consider a another right-angled triangle with sides (1,√2, √3). Here,
√3 = sec ( tan-1 (sec ( tan-1( 0! ) ) ) )
Extrapolating this idea futher, for any number x, we can represent √x using one 0 as,
√x = sec ( tan-1 (…… sec ( tan-1 (0!)) ……)), where sec ( tan-1 ….) is taken x-1 times

Sol 2 :- (((((0!)++)++)++)++)! = 120

If you are known to programming then it'll be easy to understand.
In C/C++ programming '++' operator stands for increment of 1.

= (((((0!)++)++)++)++)!
= (((((1)++)++)++)++)!
= (((2++)++)++)!
= ((3++)++)!
= (4++)!
= 5!
= 120