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Your problem specifies "point"......so my answer is correct based on that....

However.....it should specify "lattice points" [ points with integer coordinates ]

Taking this into cosideration.....then 1/3 is correct......  

You are very welcome! 

No, its not strange. This website features many different tools, such as the scientific calculator and forums, and you can use them however you want.

This problem requires integer coordinates, so it is not a geometric probability problem.
There are 9 choices for x and 5 choices for y so there are 9*5=45 possible choices for (x,y). The pairs for which x+y ≤ 4 are
(0,0) (0,1) (1,0) (0,2) (1,1) (2,0) (0,3) (1,2) (2,1) (3,0) (0,4) (1,3) (2,2) (3,1) (4,0)

These are 15 of the 45 possible choices, so the probability is 15/45, or 1/3. I still don't understand your method, could you explain it to me, please?

Mmmmm.....the area of the feasible region is  8 * 4  =  32 units^2

x + y ≤ 4  will form a triangle in this region with a base and height of 4.........so its area is 8

So....the probability of a point falling into this area   of the feasible region  =  8 / 32   =  1 / 4

Hey Stark.... see this    

https://web2.0calc.com/questions/math_6717#r1

No your image is not there.  :(

when it says P(4, why does it stop? was there more? also, what does P(4) even mean? 

The first one is right though.

It is a 4 by 4 dot grid.

Ok I'll give it another go.

Here are the options. for the 2 sides that are the same.

across 0 up 1   (1 unit)                 1 possibility as seen on circle below

across 0 up 2  (2 units)                 1 possibility as seen on circle below

across 0 up 3   (3 units)                1 possibility as seen on circle below

across 1 up 1  (sqrt2  units)           1 possibility as seen on circle below

across 1 up 2    (sqrt5 units)           3 possibilities as seen on circle below

across 1 up 3    (sqrt10 units)         2 possibilities as seen on circle below

across 2 up 2     (sqrt8 units)          1 possibility as seen on circle below

across 2 up 3      (sqrt13 units)        2 possibilities as seen on circle below

So unless I made a careless oversight (which is quite possible),  there are  12 possibilities.   

WAIT... I DONT SEE THE IMAGE I PUT

yay it works now

what do you mean by increasing and decraesing?

Agree with Melody.....I never noticed that the LCM of the first n positive integers is just the product of all the primes from 2 to n....!!!!

No histogram is shown, stark...

Well done, asdf  !!!!!

n must be even, or else the two bases in the exponent would be equal, which is impossible.

if n is even, either z or z+1/z is negative. if z is negative, then -z=z+1/z or 2z=-1/z, and 2z^2=-1 which is imposssible.

then z+1/z is negative. then -z-1/z=z, so 2z=-1/z and 2z^2=-1 which is also impossible.

so from what i get, there are no possible solutions for n such that there is a solution for z.

HOPE THIS HELPED!!

Thanks, asdf  !!!!

Here's my best attempt....

The top factors as 18 (x - 5) ( x - 2)

The bottom factors as  (x + 1) ( x - 5) ( x - 2)

So  we are left with    18 / (x + 1)

The divisors of 18 are   ± (1, 2, 3, 6, 9, 18)

This means that we have integer values when  x =   -2 , -3  -4, -7, -10, -19, 0, 1, 5, 8, 17 

Note that we cannot use x =   -1   or x = 2 because they make an original denominator = 0

So  the sum of these integer values are  - 45 + 31  =   - 14

if the radius of the small circle is x, and radius of the larger circle is y, then we are looking for x/y. y^2*pi-x^2*pi=3*x^2*pi, so y^2=4x^2, y=2x, and x/y=1/2.

HOPE THIS HELPED!!

If you become a member it will be easier for people to teach you this.

Yes we can help here.

You only need to learn a little at a time. As you learn things the pace you are learning  will automatically increase.

To make it go to a new line just add in \\  where you want the line break.  :)

I'll write a little simple code.

I want you to copy it and paste it into the latex box but also add \\  (2 backslashes) prior to the 6+8    

Then press ok   

Here is the code:

\text{coding practice please make a new line after this}6+8=14

Give it a go.

nevermind guest, i miscalculated on the question and forgot -7. it's -20!