Seating Chart

Some people believe that the easiest way to ruin a conference is to do a bad job of planning the seating arrangements. The conference's chairperson, Saanvi, is planning seating for the dinner after the keynote address, with N people, and she wants to manually review all possible seating arrangements in order to pick the absolutely best one. To figure out whether that's feasible, she's planning to write a program to compute the number of possible seating arrangements.

There are K round tables at the dinner, numbered 1 through K. It is important to have exactly the same number of people sitting at each table. If that is impossible (N is not divisible by K), then the table with the most people must have at most one more person sitting at it than the table with the fewest people.

Each of the N people will be assigned a unique number between 0 and N - 1. What matters is who is sitting next to whom, and not exactly where they're sitting. In other words, two arrangements, A and B, are considered different if there exists a pair of numbers, α and β, such that persons α and β are sitting next to each other at the same table in arrangement A, but they are not sitting next to each other in arrangement B.

For example, if N is 5, and K is 2, we must have 3 people seated at one of the tables, and 2 people seated at the other table. Here is the list of all 10 of the possible arrangements:

[[0, 1, 2], [3, 4]]
[[0, 1, 3], [2, 4]]
[[0, 1, 4], [2, 3]]
[[0, 2, 3], [1, 4]]
[[0, 2, 4], [1, 3]]
[[0, 3, 4], [1, 2]]
[[1, 2, 3], [0, 4]]
[[1, 2, 4], [0, 3]]
[[1, 3, 4], [0, 2]]
[[2, 3, 4], [0, 1]]

All other arrangements are similar to one of the arrangements above and are not counted as different. In particular, all of the following arrangements are considered to be the same:

[[0, 1, 2], [3, 4]]
[[2, 0, 1], [3, 4]]
[[1, 2, 0], [4, 3]]
[[0, 2, 1], [3, 4]]
[[3, 4], [0, 2, 1]]

This is because the following pairs of people (and no other pairs) are sitting next to each other in each of these 5 arrangements:

0 and 1
0 and 2
1 and 2
3 and 4

Another example is N = 5 and K = 3, which requires having two tables with two people each, and one table with a single person sitting at it. There are 15 possible arrangements in this case:

[[0, 1], [2, 3], [4]]
[[0, 1], [2, 4], [3]]
[[0, 1], [3, 4], [2]]
[[0, 2], [1, 3], [4]]
[[0, 2], [1, 4], [3]]
[[0, 2], [3, 4], [1]]
[[0, 3], [1, 2], [4]]
[[0, 3], [1, 4], [2]]
[[0, 3], [2, 4], [1]]
[[0, 4], [1, 2], [3]]
[[0, 4], [1, 3], [2]]
[[0, 4], [2, 3], [1]]
[[1, 2], [3, 4], [0]]
[[1, 3], [2, 4], [0]]
[[1, 4], [2, 3], [0]]

In this final example, N = 5 and K = 1, which means that we only have a single table, seating all 5 guests. Here, the answer is 12:

[[0, 1, 2, 3, 4]]
[[0, 1, 2, 4, 3]]
[[0, 1, 3, 2, 4]]
[[0, 1, 3, 4, 2]]
[[0, 1, 4, 2, 3]]
[[0, 1, 4, 3, 2]]
[[0, 2, 1, 3, 4]]
[[0, 2, 1, 4, 3]]
[[0, 2, 3, 1, 4]]
[[0, 2, 4, 1, 3]]
[[0, 3, 1, 2, 4]]
[[0, 3, 2, 1, 4]]

The first line of the input gives the number of test cases, T. T lines, each one containing two integers, N and K.

For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the number of different possible seating arrangements.

• 1 ≤ K ≤ N.