CPhill

Let ( x, y)  be  the coordinates of Q

Using the  mid-point formula  we have

x coordinate of P + x coordinate of Q 

_________________________________   =   x coordinate of  M

                     2

[ 11 + x ]        =       5           multiply both sides by 2

_______

      2

11 +  x  = 10     subtract  11 from both sides

x  =  -1

And  

 y coordinate of P  +  y coordinate of Q     =  y coordinate of M

_________________________________

                      2

[ -10  +  y  ]   =   -  2                multiply both sides by  2

___________

        2

-10 +   y  =  -4          add 10 to both sides

y   = 6

So.... Q =  ( x, y  )  =  ( -1 , 6 )

We need to know each contractor's rate per hour to answer this....

Well...what terms do give you  trouble  ???

I think this situation  is fairly difficult to evaluate....but I believe that the answer involves something  known as "Stirling Numbers of the Second Kind"

The  number of ways of distributing  6  balls into 3  indistinguishable  boxes  [  assuming that one or more boxes may be empty ]   is given by this sum :

S2 ( 6,1)   +  S2(6,2)  + S2(6,3)  =

1     +           31        +    90 =

122  ways

P.S.  - If someone knows more about this....corrections are welcome  !!!!!

Yeah.....you probably would have at some point....some of these trig equations can be real "stumpers"  !!!!

We just did one, here   :  https://web2.0calc.com/questions/test_12

Just substitute   "x"   for "N"  and you got it  !!!!!

5cos^3 (x)   =  5 cos (x)      ...... subtract 5 cos (x)   from both sides

5cos^3(x)  - 5 cos (x)    = 0        .....divide through by  5

cos^3(x) - cos (x)  =  0      ....factor

cos  (x )  ( cos^2 (x)  - 1)   = 0

We have two  equations to consider

cos (x )  =  0     and this happens at   pi /2 and 3pi/2    in the given interval

Also

cos^2 ( x)   - 1   = 0       ....add 1 to both sides

cos^2 (x )   =  1      take both roots

cos (x)  = 1            or       cos (x)  = -1

This happens at               This happens at

x = 0                                     x =      pi   

So...the solutions are

x = 0 ,   pi/2,   pi,  3pi/2

Here's graph to confirm this : https://www.desmos.com/calculator/jx5rslbflo

Rose is correct

The probability that the  second woman receives a pink rose is dependent upon what the first woman receives

If the first woman does not receive a pink rose....the probabilty of the second woman receiving one is 4/ 14 =  2 / 7

It the first woman does receive a pink rose....the probability of the second woman receiving one is  3 / 14 

We have a total of  8 + 12 + 6 + 2  =   28 students

Of these....2 are  in Group 4...so...the probaility  is

[  Number in Group 4  ] /  [ Total Number  ] =   2  / 28 =   1 / 14

Hey...we  BOTH got the  correct answer....two great minds can't possibly be wrong....LOL!!!!