Actual responses to actual questions from students of Math 344, Introduction to Combinatorics (Spring 2004).
Could the answer to part (a) be 51?
Yes, of course.
Could the answer to part (c) be 46·2 =92?
Explanation: x can be any number from 0,1,2,3,4, all the way to 45, and the corresponding y value is from 5,6,7,8,9 all the way to 50 (since y has to be 5 greater than x). Then multiply by two since it is an unordered pair (x,y) ≠ (y,x).
No, the correct answer is 46. The problem with your solution is that unordered means (x,y) = (y,x). That's why there are only 46.
I've reasoned as follows: We have 10 slots (length 10), and so:
1st slot: 3 possibilities out of (0,1,2)
2nd slot: 2
3rd slot: 2
4th slot: 2
5th slot: 2
6th slot: 2
7th slot: 2
8th slot: 2
9th slot: 2
10th slot:2
This gives us a total of 3⋅29 possibilities.
Am I right?
Yes, I believe so.