[Math] A few basic Counting Problems

Question

I don't know if I got these correct. Can someone check for me?

1. How many ways are there to roll a sum of 7 with three standard 6-faced die?
   There is:
   1,1,5
   1,2,4
   1,3,3
   1.4.2
   1,5,1
   2,1,4
   2,2,3
   2,3,2
   2,3,1
   3,1,3
   3,2,2
   3,3,1
   4,1,2
   4,2,1
   5,1,1
   15????

2. Uncle Henry has 10 one dollar bills to distribute to his 5 nieces and nephews. How many ways are there to distribute the money?
He can give 10 to one niece/nephew and 10 to the others, 5 ways to do this.
He can give 9 to one niece/nephew, 1 to another, and 0 to the rest. 20 ways to do this.
He can give 8 to one niece/nephew, 1 to 2 different nieces/nephews, and 0 to the rest. 60 ways to do this.
He can give 8 to one niece/nephew,2 to another, and 0 to the rest 20 ways to do this.

I kept doing this and finally got 6825

3. In how many ways can three identical rattles be given to two different babies?
   0,3
   1,2
   2,1
   3,0
   4?

4. A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this polygon have?
   I'm lost on this one.

5. Each of the numbers 1 through 10 are placed in a bag and drawn at random with replacement. How many ways can three numbers be drawn whose sum is 13?
   1,2,10
   1,3,9
   1,4,8
   1,5,7
   1,6,6
   1,7,5
   1,8,4
   1,9,3
   1,10,2
   9 ways for this, then 8,7,6,5,4,3,2,1 for the other ones. 45?
   

6. How many diagonals does a seven sided regualar polygon have?
   
   Also lost on this

7. Less that 50 people are at a party. Each person shakes everyone else's hand. IF there is an odd number of total handshakes at the party, what is that largest number of people that could be at the party?
   49?
   
   That's the largest perfect square less than 50…

And thats it

Answer 1

For 6) note that the number of diagonal for a $n$ sided polygon is $\frac{n(n-3)}{2}$.