Problems on combinatorics The following comes from questions comes from a recent combinatorics paper I attended :
1.27 people are to travel by a bus which can carry 12 inside and 15 outside. In how many ways can the party be distributed between inside and outside if 5 people refuse to go outside and 6 will not go inside?
The solution given C(16,7), I have no clue how they got it ?!
2.The number of functions f from the set A = {0, 1, 2} into the set B = {1, 2, 3, 4, 5, 6, 7} such that $f(i) \le f(j) $ for $i \lt j $ and $i,j$ belongs to A is
The solution given is C(8,3). I didn't really understood this one.
3.The number of ordered pairs $(m, n) m, n $ is in {1 , 2, … , 100} such that $7^m + 7^n$ is divisible by 5 is
The solution given is 2500, but how ?
4.The coefficient of $x^{20}$in the expansion of $(1 + 3x + 3x^2 + x^3)^{20}$, is ?
How to solve this one elegantly ? 
5.An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, …, 9 without replacement. The number of ways in which this can be done is:
Now this one seems impossible for me to solve in 1 mint,or is it ? Given soln is 36(7!)

For number 4 you would use the multinomial theorem.
