CPhill

I'll have a go at this one.....!!!

Since the perimeter is 64, the width and the length must be (1/2) of this = 32

And if each is an integer quantity....we have the following possibilites

31 + 1  ... 30+  2 ...  29 + 3....etc.

But notice, when we get to 16 + 16, we can stop....because all combinations after this are just "reversals" of the sums that came before 16 + 16 !!!

So

31 - 16 + 1   = 16  noncongruent rectangles.....!!!!

Thanks.....I get lucky every now and then on these "counting" problems..........LOL!!!

We can choose 1 of the 3 triplets and from the other 11 players, any 5 of them.

So......C(3,1) * C(11,5)  = 1386 different teams...

I'll give this one a try  ....though it doesn't appear to be too easy!!!

Notice that we could have these 2 configurations right off the bat......

yyyy zzzz xxxx      and  zzzz xxxx yyyy

Next, we could have this

yyyz  xzzz xxxy  .......but....since each single letter within each grouping can appear  in one of four positions,

we have 4 x 4 x 4 = 64 possiblities

Next, we could have

yyzz xxzz xxyy ....notice that within each subgroup, we have (4!)/(2! * 2!)  = 6 recognizable "words"

So  6 x 6 x 6 = 216 possibilities

We could also have

yzzz xxxz xyyy...which is again, 64 possibilities

So...if I haven't missed any combos, the total possible "words" = 2 + 2(64) + 216 = 346

Could somebody else check this....(Melody, Alan, heureka ??)

Since 5 evenly divides 60, it already IS to the "nearest 5m^2."

Alan...in the first one, don't (±5, 0) and (0,±5) also "work ??"

And in the second....don't  (±7, 0, 0), (0,±7, 0 ) and (0, 0, ±7) also do the job??

Well since "C" is in a given position, and we're going in a clock-wise manner, we have 4 ways to choose the next letter, 3 ways to choose the letter after that, etc.

So  4 x 3 x 2 x 1= 4!  = 24 arrangements

OK...post the questions, and we'll see what we can do....!!!!

Using the Law of Sines, we have

31 / sin X  = Z / sin Z

31 / (2/3)  = Z / (1/4)  simplify

93/2 = 4Z     divide both sides by 4

93/8 = Z = 11.625