The time complexity of answer is O(m*n*N).

And we need to use **dynamic programming (for remembering the existing searching)** and **DFS (for searching the answer)** to solve this problem.

C++ version:

```
class Solution {
private:
int m, n, N, dx[4] = {-1, 1, 0, 0}, dy[4] = {0, 0, -1, 1};
const int mod = 1e9 + 7;
bool check(int i, int j) {return i >= m || j >= n || i < 0 || j < 0;}
public:
int findPaths(int m, int n, int N, int i, int j) {
this->m = m, this->n = n, this->N = N;
vector<vector<int>> dp(m * n, vector<int>(N, -1));
return solve(i, j, N, dp, 0);
}
int solve(int i, int j, int step, vector<vector<int>>& dp, int ans) {
if (check(i, j)) return 1; // out of boundary, count as 1 way
if (step == 0) return 0; // without steps but not out of bounday, don't count as a way
if (dp[i*n + j][step-1] == -1) {
// the answer came from 4 paths: top, down, left, right
for (int k=0; k<4; ++k)
ans = (ans + solve(i + dx[k], j + dy[k], step-1, dp, 0) % mod) % mod;
dp[i*n + j][step-1] = ans;
}
return dp[i*n + j][step-1];
}
};
```

Python version:

```
class Solution(object):
dx = [-1,1,0,0]
dy = [0,0,-1,1]
lc = 1e9 + 7
def solve(self, i, j, step, dp, ans, m, n, N):
if i >= m or j >= n or i < 0 or j < 0:
return 1
if step == 0:
return 0
if dp[i*n +j][step - 1] == -1:
for k in xrange(4):
ans = (ans + self.solve(i + self.dx[k], j + self.dy[k], step-1, dp, 0, m, n, N) % self.lc) % self.lc
dp[i*n + j][step - 1] = ans
return int(dp[i*n + j][step - 1])
def findPaths(self, m, n, N, i, j):
dp = [[-1 for t in xrange(N)] for k in xrange(m*n)]
return self.solve(i, j, N, dp, 0, m, n, N)
```