CPhill

The hour hand moves  at  1/2°  every minute

The minute hand moves at  6°  every minute

Using  the top  of the clock  as 0°.....at   12:25  

The  minute hand has moved  25 * 6  =  150° from the top of the clock

The hour hand has moved   25 * 1/2  = 25/2  =  12.5°  from the top of the clock

So.....the angle  between them is     150 - 12.5  =  137.5°

We can use the "formula" for the distance from a point to a line to solve this

The line  is     2x + y -1   = 0

So we have that

l  2(3) + 4 - 1 l            l  9  l             9

____________  =     ____   =      ___     this is the radius of the circle

 √[2^2 + 1^2 ]             √5               √5

The equation of the circle is

(x - 3)^2  +  ( y - 4)^2  =   81/5

Here  is a graph  : https://www.desmos.com/calculator/b8fxaseqdv

6!  +  4!

_______        Not  as hard as it seems.....we can split this up as.....

     5!   

6!           4!

__   +   ___    =

5!           5!

6 * 5 * 4 * 3 * 2 * 1            4 * 3* 2 * 1

_______________  +     ____________   =

5 * 4 * 3 * 2 * 1                5* 4 * 3 * 2 * 1

              1

 6   +     __

              5 

Just as in your other question  with distiguishable balls and identical boxes, we have

S₂ (5, 1)  +  S₂ (5, 2)  + S₂ (5, 3)   =

     1        +     15         +      25      =

                     41   ways  

OK....!!!

Mmmmmm....I'm still trying to figure that out......

This involves something  known as  a  "Stirling Number of the Second Kind"

The total number  of  ways  =

S₂ ( 4, 1)  +  S₂ (4,2)  + S₂ (4, 3)  =

      1      +        7         +        6    =

              14 ways

Pretty crafty, Melody......!!!

I'm going to look at this in more depth to make sure I understand it fully  !!!

Box 1  Box 2

  5         0

  4         1

  3         2

  2         3

  1         4

  0         5

This can be verified  by the  formula

C ( k + n - 1 , n - 1)     where  k  is the number of balls  and n  is the number of boxes

So

C ( 5 + 2 - 1, 2 - 1)  =  C ( 6 , 1)  =  6  ways

3 Cylinders  =  5 Spheres    (1) 

3 Spheres = 12 Cubes     (2)

Multiply (2)  by 5  on each side

60 Cubes  = 15 Spheres   (3)

Multiply (1) by 3 on each side

9 Cylinders =  15 Spheres  (4)

So   (3)  and (4)  imply that

9 Cylinders =  60 Cubes