Score : $1100$ points

Problem Statement

Niwango bought a piece of land that can be represented as a half-open interval $[0, X)$.

Niwango will lay out $N$ vinyl sheets on this land. The sheets are numbered $1,2, \ldots, N$, and they are distinguishable. For Sheet $i$, he can choose an integer $j$ such that $0 \leq j \leq X - L_i$ and cover $[j, j + L_i)$ with this sheet.

Find the number of ways to cover the land with the sheets such that no point in $[0, X)$ remains uncovered, modulo $(10^9+7)$. We consider two ways to cover the land different if and only if there is an integer $i$ $(1 \leq i \leq N)$ such that the region covered by Sheet $i$ is different.

Constraints

• $1 \leq N \leq 100$
• $1 \leq L_i \leq X \leq 500$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $X$

$L_1$ $L_2$ $\ldots$ $L_N$

Output

Print the answer.

3 3
1 1 2

10

• If we ignore whether the whole interval is covered, there are $18$ ways to lay out the sheets.
• Among them, there are $4$ ways that leave $[0, 1)$ uncovered, and $4$ ways that leave $[2, 3)$ uncovered.
• Each of the other ways covers the whole interval $[0,3)$, so the answer is $10$.

18 477
324 31 27 227 9 21 41 29 50 34 2 362 92 11 13 17 183 119

134796357

• Find the number of ways modulo $(10^9+7)$.