igotchu

We know a rectangle has two congruent pairs of sides.

So, we are trying to find what multiplies to 20 and adds to 9 (half of 18 because we are only counting one pair).

x*y = 20

x + y = 9

The factors of 20 are 1,2,4,5,10,20 and two of those add up to 9.

When we look carefully we find those two are 4 and 5.

Sooo the dimensions of the rectangle are 4 and 5.

(Tell me if anything is wrong here) 

I'm gonna take a guess here and say that your diagram meant to look like this:

        A        

     RRR

   CCCCC

 HHHHHHH

This can be done with a simple counting principle:

( I actually don't know how to do this that well so I'm taking a wild guess)

From the first to the second we have 3 possible ways.

From the second to the third we also have 3 possible ways.

From the third to the fourth we again have 3 possible ways.

SO, my wild guess says the answer is 27 (but it was a wild guess so check it.

Easy...

If we let the smallest integer = x then the sum of all the integers is:

x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6 = 6(x+6)

Simplifying we get:

7x + 21 = 6x + 36

x = 15

So the smallest integer is 15.

(Please tell me if this answer is incorrect) 

Agreed there's no diagram to this.

Easy. 

When wanting to choose something from a group we use the choose function which states:

n!/r! (n-r)!

So applying this to your problem we get 16!/5!*11!

This simplifies to (16*15*14*13*12) / (5*4*3*2*1)

So the answer should be 4368 if I'm not mistaken.

Is there supposed to be a diagram for this?