C |
Cheerleaders |
In
most professional sporting events, cheerleaders play a major role in
entertaining the spectators. Their roles are substantial during breaks and
prior to start of play. The world cup soccer is no exception. Usually the
cheerleaders form a group and perform at the centre of the field. In addition
to this group, some of them are placed outside the side line so they are closer
to the spectators. The organizers would like to ensure that at least one
cheerleader is located on each of the four sides. For this problem, we will
model the playing ground as an M*N
rectangular grid. The constraints for placing cheerleaders are described below:
§ There should be at least one cheerleader on
each of the four sides. Note that, placing a cheerleader on a corner cell would
cover two sides simultaneously.
§ There can be at most one cheerleader in a
cell.
§ All the cheerleaders available must be
assigned to a cell. That is, none of them can be left out.
The
organizers would like to know, how many ways they can place the cheerleaders
while maintaining the above constraints. Two placements are different, if there
is at least one cell which contains a cheerleader in one of the placement but
not in the other.
Input
The
first line of input contains a positive integer T<=50, which denotes the number of test cases. T lines then follow each describing one
test case. Each case consists of three nonnegative integers, 2<=M, N<=20 and K<=500. Here M is the number of rows and N
is the number of columns in the grid. K
denotes the number of cheerleaders that must be assigned to the cells in the
grid.
Output
For each case of input, there will be one line of
output. It will first contain the case number followed by the number of ways to
place the cheerleaders as described earlier. Look at the sample output for
exact formatting. Note that, the numbers can be arbitrarily large. Therefore
you must output the answers modulo 1000007.
Sample Input |
Sample Output |
2 2
2 1 2
3 2 |
Case 1: 0 Case 2: 2 |