**Q24.**

Denote as the *i-*th bucket that contains the puck.

If all 4 pucks are distributed into the same Green bucket, then we don’t need to worry about the Red, Blue, and Yellow buckets since they won’t have 4 pucks in them. So,

Now, 3 pucks are distributed into the same Green bucket, then we only need to exclude the case where 3 pucks are put into the same Red bucket. There are 4 ways to permute *aaab* and 3 numbers to pick for *a* and *b* which is 6 ways, so

If 2 pucks are distributed into the same Green bucket, then we only need to consider the case where all 3 pucks are distributed evenly into three different Red buckets and all 2 pucks are distributed evenly into two different Blue buckets. The most difficult part of the problem is in this case here. We need to find the number of ways to permute *n* repeated objects, *aabc*, which is , and since the permutation of *b* and *c* has already counted, we need to only consider how many numbers go into the choice of *a* which is 3.

Final step, add the three results and we get

**Q25.**

For and

When *k* = 1,

. The 8 quadruples are easy to find, they are

When *k* = 2,

When *k* = 3,

or

Therefore, *R* can only be 3, 6, or 8.

When *R* = 3,

Therefore 2017 quadruples.

When *R* = 6,

Therefore 2017 quadruples.

When *R* = 8,

Now, obviously has no solution.

If *k* = 2,

is a solution.

In conclusion, *N* = 8 + 2017 + 2017 + 1 = 4043.

Therefore the sum of digits of *N* is 11.