CPhill

Here's the third one

From any of the 13  ranks we want to choose  any 2 of them

And from each rank we want to choose and 2 of 4 cards

And from the other 44 remaining cards, we want to choose 1 of them

So....the total number of pairs are

C(13,2) * C(4,2)^2 *  C(44,1)  =  123,552

Here's the second one

First, just choose any one of the colors to occupy any two opposite sides

Then, the other 3 colors can be arranged in 3!  =  6 ways

So......

6 different patterns are possible

Here's the first one

Anchor the British Diplomat in any position

The Germans can then be arranged in 4! ways = 24 ways  and the French can be seated in 3! ways = 6 ways

And the Germans and French themselves can each be seated in 2! ways  = 2 ways  on either side of the British  diplomat....so the total arrangements  =  2! * 3! * 4!  =  2 * 6 * 24  =  288 arrangements

Cheko has been saving part of his allowance in a bag every month. The bag containing $2 and $10 notes had a total value of $1116. 6 pieces of $10 notes were exchanged for $2 notes of the same value. In the end, the number of $10 notes was the same as the $2 notes. How many $2 notes were there at first?

Call the number of $2 and $10 notes at the end, N

So

2N  +  10N  =  1116    simplify

12N  = 1116     divide both sides by 12

N =  93

And 6 $10 bills  =  $60

So.....this = 30  $2  bills

So....there must have been  93 - 30  =  63  $2 bills initially  and 99 $10 bills

Check

2(63)  + 10(99)  =  $1116

l 3x + 1 l  +  2  =  0

Notice that the result of the absolute value will be ≥ 0.....so  adding 2 to this will be positive......thus.....the left side can never  = 0

Taylor Series  around x = 0

x + x^3/6 + (3 x^5)/40 + (5 x^7)/112 + (35 x^9)/1152 + (63 x^11)/2816 

y=2.6x

y=0.5558x^2-2.482x+4.4573

Set the y's equal.....and we have

2.6x  =  0.5558x^2-2.482x+4.4573     subtract  2.6x from borh sides

 0.5558x^2 - 5.082x +4.4573  =  0

And the solutions are  

x ≈  0.982689      and   x  ≈   8.16089

We can see right away that there is no solution....it's impossible to subtract a non-zero quantity from something - in this case, 4m - and have the result equal to 4m

Odds against  = [ 7 - 4] :  4   =  3 :  4

Odds against  =  [25  - 1 ] : 1   =   24 :  1