DerangementsStrikeBack | Topcoder Archive

Problem Statement

You are given two ints: n and m.

Let D[i] be the number of permutations of the set {1,2,...,n+i} such that the first i values are not fixed points of the permutation. Formally, we are interested in permutations p such that for each j between 1 and i, inclusive, we have p(j) != j.

Next, let B[i] = (D[i] / n!) modulo (10^9 + 7). Note that it can be shown that D[i] divided by n! is always an integer.

Compute and return the bitwise xor of the values B[i] for i between 1 and m, inclusive.

Definition

Class:: DerangementsStrikeBack
Method:: count
Parameters:: int, int
Returns:: int
Method signature:: int count(int n, int m)
(be sure your method is public)

Notes

Author's solution explicitly computes each of the values B[i]. Bitwise xor is only used to reduce the size of the output.

Constraints

n will be between 0 and 1,000,000,000, inclusive.
m will be between 1 and 100,000, inclusive.

Examples

```
0
```
```
3
```
```
Returns: 3
```
As we have n=0, we are counting derangements, i.e., permutations with no fixed points. We can easily see that D[1]=0, D[2]=1, and D[3]=2. As 0! = 1, we also have B[1]=0, B[2]=1, and B[3]=2. Thus, the correct return value is (0 xor 1 xor 2) = 3.
```
1
```
```
3
```
```
Returns: 9
```
```
3
```
```
3
```
```
Returns: 73
```
```
4
```
```
1
```
```
Returns: 4
```
```
70
```
```
39
```
```
Returns: 319676671
```

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