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1/ (a-2)  +  1 / a   = 1 / (2 2/3)        solve for a

3 c 1.5 d  / 3 e =    1.5 chicken days per egg

200 egg  * 1.5 c - d /egg    / 2 c   = 150 days  

The probability is =4! / [3!.1!.2^4] =24 /[6 x 16] = 1/4

Disagree

Now set k=1, k=2, k=3 one after the other. Because of x^3
you get 3 solutions

AAABBBBBBB =10! / 3!.7! =120 permutations

When a permutation begins with the letter "A", there will be: 3 x 7 =21 permutations where no 2 A's will be adjacent to each other.

When a permutation begins with the letter "B", the first permutation will begin with:BAAABBBBBB, and each A will move 5 spots away from each other for a total of:5 x 7 "B's" =35 permutations where no 2 A's will be adjacent to each other.

So, the total = 21 + 35 = 56 permutations out of 120 where no 2 A's will be adjacent to each other.

"this is the best explanation I can give"   Not just your best, I think it's the best anybody could give, simple and elegant.  Or is simplicity already an element of elegance?  Anyway, good work!  I was attempting a solution based on the ratio of the circumference of the circle being 2π radii.  I quickly bogged down, and then realized I was completely on the wrong tack. 

_.

Done!

A point of tangent forms a 90^o angle with the radius or diameter to that point.

An inscribed angle's measure is 1/2 of the arc it intercepts  BCA intercepts an arc of 180^o   so BCA is 90^o  too.

Hint so that you can still maybe get it: Connect the two random points with the center, then think about when the chord is equal/larger/smaller than the radius.

answer is here \/

If the two points make a 60-degree angle with the center, then the chord will be equal to the radius (because the three points form an equilateral triangle). 

It doesn't matter where we put the first point, but it does matter where we put the second point. If the second point is within 60 degrees of the first angle, then the chord will be shorter than the radius. So the arc of the circle where we can put the second point has measure 120 degrees (the 60 degrees goes on either side of the point), with a 1/3 chance of putting the point there. To find the probability that the chord is longer, just subtract 1/3(chance of the chord being shorter) from 1 to get \(\boxed{2/3}\) .

A diagram would be kinda useful but this is the best explanation I can give.

21 PI  is a fraction of the total 49 pi

21pi / 49 pi  x 360^o = .........^o

Hmmmm       16 possibilities for the coin toss

there are 4 possibles for EXACTLY 3 heads

HHHT

HHTH

HTHH

THHH       4 OUT OF 16 =   1/4