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arcsin   [ (1017.5) * sin (145) / 2035 ]

Paste this into this website  and hit "enter"

https://www.wolframalpha.com/input/?i=arcsin%C2%A0+%C2%A0%5B+(1017.5)+*+sin+(145)+%2F+2035+%5D

I'm using   x for a and y for b

Look at the graph, here :  https://www.desmos.com/calculator/5iayh796gi

The  region formed by the first two inequalities is 24 units^2

The region formed by ab > 0 ⇒ xy > 0    is split into two rectangles

One is 3 x 2   = 6 units^2

And the other is 1 x 4 = 4 units^2

So....the probability that   ab > 0   ⇒  xy > 0  is  

[ 6 + 4 ] / 24    =  10 / 24    =    5 / 12

cos (theta)  = -2 / √5

sin theta =  1/√5

tan (theta + pi/4)   =

sin (theta + pi/4)

_______________  =

cos (theta + pi /4)

sin theta * cos pi/4   + cos theta * sin pi/4

_________________________________ =         

costheta * cos pi/4 -  sin theta * sin pi/4

 { sin pi/4 = cos pi/4 =  1 /√2 }

1/√2  [ sin theta + cos theta ]

_______________________  =

1 / √2 [ cos theta - sin theta ]

[ 1/√5  + (-2)/√5 ]

_______________   =

[ (-2)/√5  - 1/√5 ]  

[ 1 - 2 ]                  -1                 1 

________    =      ____     =    ___

[ -2 - 1 ]                  -3                3

See this graph : https://www.desmos.com/calculator/3psnr4vmhw

The area covered by the first two inequalities is   18 units^2

The  inequality    x + y ≤ 4   forms a trapezoid inside this region

Its   area   is   (1/2)(3) ( 4 + 1)    =  15/2   units^2

So....the probability that a point is chosen inside the trapezoid  in the feasible region is

(15/2) / 18   =    15/ 36      =  5 / 12

Let the side of the cube be, S

HA  = S√2

AC = S√2

CH = S√2

So.....HAC forms an equilateral triangle.....and each angle = 60°

So....the sin 60°   = sin HAC =   √3 / 2 

Nvm i got it, Thanks anyways Cphill

When it is stationary, v = 0

If we take the derivative of the displacement fxn , it is the velocity function

   then we can set the VELOCITY function = 0 to find TIME to substitute in to the DISPLACEMENT fxn

the derivative is

    6t-12 = velocity

    6t-12 = 0   yields t = 2 sec

Substitute this in to the displacement fxn

3(2^2) -12(2) +15 = 3 m

Maybe.....ya have more info?  

It's not me downvoting you. Which Guest are you, Guest #1, Guest #2, or a guest I haven't met yet?

why are you downvoting me tho

I just don't understand it well.

wait, why is this downvoted? I thought he knew what n choose k means because he used it before 

https://web2.0calc.com/questions/strange-q#r5

I thought you know what n choose k means

I do take AoPS, but it's only the beginning of the year. Counting and Probability is the last 10 weeks of class.

I just got lucky.....LOL!!!!

Thank you, CPhill, for providing a wonderful solution! I was actually stumped at first..

Thanks, tertre     [ and tanmai ]   !!!  

Oh yes, there is! Here: \(\frac{n!}{r!(n-r)!}\)- combinations; order does not matter

The order does matter( Permutations): \(\frac{n!}{(n-r)!}\)

If you take AoPS, they give you wonderful examples! 

So is there, like, a formula or something to compute that?

Also, a quote from CPhill's explanation: "We have   C(16, 4)   = 1820   possible sets of any four of the points."

We can express this as \(C\binom{16}{3}\) , where it is combinations. 

It's not a fraction. However, it means 16 choose 4; it computes the number of ways to choose items from a collection of items. 

You got it right!

I've got another question, though:

\(${16 \choose 4}$\)

What does that mean? There is no line to symbolize that it is a fraction.

Here's my best attempt, tanmai....

o       o         o         o

o      o         o          o

o      o         o          o 

o      o         o          o   

We have   C(16, 4)   = 1820   possible sets of any four of the points

But.....the only ones that will be collinear points  are the  four horizontal and vertical patterns and the two diagonal patterns   =   10 total

So....the probability   is      10 / 1820   =   1 / 182