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P(both red)  =  (2/15)(1/14)  = 2/210  =  1/105

This is from the AoPs intro to counting and probability class...

WE ARE NOT HERE TO DO YOUR WORK FOR YOU...

That's an Art of Problem Solving question from Intro to Algebra... I took that...

DON'T ASK AOPS QUESTIONS

We are not here to do your work for you...

There are 2 ways we can attack this problem...

Method 1:

Notice that the subtraction result for every 2 pairs has the answer of 2.

Since there are 50 pairs that have the result of 2 this is 50(2) or 100.

Method 2:

We can use Gaussean Addition...

The normally the formula we would use is n(n+1)/2 but we are adding all the added pairs, all the subtracted pairs and then subtracting them (if that made any sense...). Since 99+3 = 102 we multiply that by 50/2 or 25. 102 x 25 = 2550.

Each pair from the subtracting side sums up to 98. We multiply 98 x 25 to get 2450. 2550 - 2450 = 100.

100 is my final answer...

100000000% agree!

Thank You CPhill for helping people on this forum site everyday...

We deeply appreciate it...

a + b  = 1             

Square both sides  of this

a^2 + 2ab + b^2   = 1

(a^2 +b^2) + 2ab   = 1

  (2)  + 2ab = 1

      2ab   =  -1

        ab  = -1/2

        -ab  =1/2

a^3 + b^3   =  (a + b) (a^2  - ab + b^2)  =  (a + b)  [ (a^2 + b^2 ) + - ab)  =  (1) [ 2 +1/2] =  5/2

yeah i'm confused about this too... 

You copied the AoPS solution.

Thx!

This image is excellent I recommend to view it!

https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.onlinemath4all.com%2Fgemdas-order-of-operations.html&psig=AOvVaw3v8y-v4md6S_WjSpIc2j7Q&ust=1588636312946000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCODHq8vxmOkCFQAAAAAdAAAAABAD

Excellent, LD!!!!!!

Thanks, Ducky! but what about factorials and brackets how do they fit in?

-2⁴  = 16

- 2^4 =  - 16

-3^3 = -27

-3^3 =  - 27

Ok here's my solution, 

\(\begin{align*} (-2)^4 &= (-2)(-2)(-2)(-2) = 16,\\ 2^4 &= 2\cdot 2\cdot 2\cdot 2 = 16,\\ (-3)^3&= (-3)(-3)(-3) = (9)(-3) = -27,\\ 3^3 &= 3\cdot3\cdot3 = 27, \end{align*}\)

\((-2)^4 -2^4 + (-3)^3 - 3^3 = 16-16 -27-27 = \boxed{-54}.]\)

So the thing to take away in this question is to always remember that in solving these questions the order of operations in KEY!!!

We have a vertical asymptote  at  x  = 5

This  means  that  -2(5)  + c  = 0      so   c = 10

Since we  have   a horizontal asymptote at  y  =4.....this means that  (a/-2)  = 4   so  a = -8

Also....it appears that the  point  (0,0)  is  on the graph

So    -8(0)  + b =  0     means that  b  = 0

So

a + b +  c   =  -8 + 0 + 10   =    2

Oh thats cool! 

HAHAHA!!!......nope.....I'm retired......I just do this to help others.....(when I can)

990° =   720°  +  270°  =   360°  + 360°   + 270°

There will be   two occurences  of this in every  360°   and  one  more between 0 and 270°

So....there will be 2 + 2 + 1  =   5 values of  x

But like don't you have a job and stuff?

I take it by spells......some days I'm not on here, at all.....somedays, I'm feeling more "math-y".....LOL!!!

I noticed that to LOL!

-16t^2  + 20t  + 9

Te time to reach the max height  can be found as

-20 / [ 2 * -16]  =   -20/ -32  =   5/8 sec

Putting this back into the function we find the max height as

-16(5/8)^2  + 20(5/8)  + 9  =

-16 ( 25/64) + 100/8  + 9  =

-400/64  + 100/8  + 9  =

-100/16 + 200/16 + 144/16 =   244/16  = 15.25 ft

Thank you!

Find the midpoint  of  (-2,0)  and (2,0)   =  (0,0)

The  line perpendicular to this point is just the x axis and has the equation x  = 0

Find the  midpoint  of  (2,0)  and(3,2)   = [ 3+2/2, 2+0/2)  = (5/2, 1)

The slope of the  line  through these points  = [ 2-0]/[3 - 2] =  2

And the slope of a perpendicular line passing through the  midpoint we just found has the slope -1/2

The equation of this line is

y = (-1/2) ( x - 5/2) + 1

Letting x  = 0  we can find the  y coordinate of the  center of the  circle

y = (-1/2)(0 -5/2) + 1  = 5/4 + 1  = 9/4

So...the center of the circle is  (0, 9/4) 

And  the square of the radius is  2^2   + (9/4)^2  =  4+ 81/16  = 145/16

Here's the graph  https://www.desmos.com/calculator/wdbs8u2edh