Problem Statement

Snuke has N integers: 1,2,\ldots,N. He will choose K of them and give those to Takahashi.

How many ways are there to choose K consecutive integers?

Constraints

• All values in input are integers.
• 1 \leq K \leq N \leq 50

Sample Input 1

3 2

Sample Output 1

2

There are two ways to choose two consecutive integers: (1,2) and (2,3).

Sample Input 2

13 3

Sample Output 2

11

Problem Statement

Snuke has come to a store that sells boxes containing balls. The store sells the following three kinds of boxes:

• Red boxes, each containing R red balls
• Green boxes, each containing G green balls
• Blue boxes, each containing B blue balls

Snuke wants to get a total of exactly N balls by buying r red boxes, g green boxes and b blue boxes. How many triples of non-negative integers (r,g,b) achieve this?

Constraints

• All values in input are integers.
• 1 \leq R,G,B,N \leq 3000

Sample Input 1

1 2 3 4

Sample Output 1

4

Four triples achieve the objective, as follows:

• (4,0,0)
• (2,1,0)
• (1,0,1)
• (0,2,0)

Sample Input 2

13 1 4 3000

Sample Output 2

87058

Problem Statement

Snuke has N strings. The i-th string is s_i.

Let us concatenate these strings into one string after arranging them in some order. Find the maximum possible number of occurrences of AB in the resulting string.

Constraints

• 1 \leq N \leq 10^{4}
• 2 \leq |s_i| \leq 10
• s_i consists of uppercase English letters.

Sample Input 1

3
ABCA
XBAZ
BAD

Sample Output 1

2

For example, if we concatenate ABCA, BAD and XBAZ in this order, the resulting string ABCABADXBAZ has two occurrences of AB.

Sample Input 2

9
BEWPVCRWH
ZZNQYIJX
BAVREA
PA
HJMYITEOX
BCJHMRMNK
BP
QVFABZ
PRGKSPUNA

Sample Output 2

4

Sample Input 3

7
RABYBBE
JOZ
BMHQUVA
BPA
ISU
MCMABAOBHZ
SZMEHMA

Sample Output 3

4

Problem Statement

Snuke received a positive integer N from Takahashi. A positive integer m is called a favorite number when the following condition is satisfied:

• The quotient and remainder of N divided by m are equal, that is, \lfloor \frac{N}{m} \rfloor = N \bmod m holds.

Find all favorite numbers and print the sum of those.

Constraints

• All values in input are integers.
• 1 \leq N \leq 10^{12}

Sample Input 1

8

Sample Output 1

10

There are two favorite numbers: 3 and 7. Print the sum of these, 10.

Sample Input 2

1000000000000

Sample Output 2

2499686339916

Watch out for overflow.

Problem Statement

The beauty of a sequence a of length n is defined as a_1 \oplus \cdots \oplus a_n, where \oplus denotes the bitwise exclusive or (XOR).

You are given a sequence A of length N. Snuke will insert zero or more partitions in A to divide it into some number of non-empty contiguous subsequences.

There are 2^{N-1} possible ways to insert partitions. How many of them divide A into sequences whose beauties are all equal? Find this count modulo 10^{9}+7.

Constraints

• All values in input are integers.
• 1 \leq N \leq 5 \times 10^5
• 0 \leq A_i < 2^{20}

Sample Input 1

3
1 2 3

Sample Output 1

3

Four ways of dividing A shown below satisfy the condition. The condition is not satisfied only if A is divided into (1),(2),(3).

• (1,2,3)
• (1),(2,3)
• (1,2),(3)

Sample Input 2

3
1 2 2

Sample Output 2

1

Sample Input 3

32
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Sample Output 3

147483634

Find the count modulo 10^{9}+7.

Sample Input 4

24
1 2 5 3 3 6 1 1 8 8 0 3 3 4 6 6 4 0 7 2 5 4 6 2

Sample Output 4

292

Problem Statement

You are given a simple connected undirected graph G consisting of N vertices and M edges. The vertices are numbered 1 to N, and the edges are numbered 1 to M.

Edge i connects Vertex a_i and b_i bidirectionally. It is guaranteed that the subgraph consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a spanning tree of G.

An allocation of weights to the edges is called a good allocation when the tree consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a minimum spanning tree of G.

There are M! ways to allocate the edges distinct integer weights between 1 and M. For each good allocation among those, find the total weight of the edges in the minimum spanning tree, and print the sum of those total weights modulo 10^{9}+7.

Constraints

• All values in input are integers.
• 2 \leq N \leq 20
• N-1 \leq M \leq N(N-1)/2
• 1 \leq a_i, b_i \leq N
• G does not have self-loops or multiple edges.
• The subgraph consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a spanning tree of G.

Sample Input 1

3 3
1 2
2 3
1 3

Sample Output 1

6

An allocation is good only if Edge 3 has the weight 3. For these good allocations, the total weight of the edges in the minimum spanning tree is 3, and there are two good allocations, so the answer is 6.

Sample Input 2

4 4
1 2
3 2
3 4
1 3

Sample Output 2

50

Sample Input 3

15 28
10 7
5 9
2 13
2 14
6 1
5 12
2 10
3 9
10 15
11 12
12 6
2 12
12 8
4 10
15 3
13 14
1 15
15 12
4 14
1 7
5 11
7 13
9 10
2 7
1 9
5 6
12 14
5 2

Sample Output 3

657573092

Print the sum of those total weights modulo 10^{9}+7.