Contest Duration: - (local time) (100 minutes) Back to Home
H - Random Robots /

Time Limit: 2 sec / Memory Limit: 1024 MB

Score : 600 points

### Problem Statement

There are K robots on a number line. The i-th robot (1 \leq i \leq K) is initially at the coordinate x_i.

The following procedure is going to take place exactly N times.

• Each robot chooses to move or not with probability \frac{1}{2} each. The robots that move will simultaneously go the distance of 1 in the positive direction, and the other robots will remain still.

Here, all probabilistic decisions are independent.

Find the probability that no two robots meet, that is, there are never two or more robots at the same coordinate at the same time throughout the procedures, modulo 998244353 (see Notes).

### Notes

It can be proved that the probability in question is always a rational number. Additionally, under the Constraints in this problem, when that value is represented as \frac{P}{Q} using two coprime integers P and Q, it can be proved that there uniquely exists an integer R such that R \times Q \equiv P\pmod{998244353} and 0 \leq R \lt 998244353. Find this R.

### Constraints

• 2 \leq K \leq 10
• 1 \leq N \leq 1000
• 0 \leq x_1 \lt x_2 \lt \cdots \lt x_K \leq 1000
• All values in input are integers.

### Input

Input is given from Standard Input in the following format:

K N
x_1 x_2 \ldots x_K

2 2
1 2

### Sample Output 1

374341633

The probability in question is \frac{5}{8}.

We have 374341633 \times 8 \equiv 5\pmod{998244353}, so you should print 374341633.

2 2
10 100

### Sample Output 2

1

The probability in question may be 1.

### Sample Input 3

10 832
73 160 221 340 447 574 720 742 782 970

553220346

### 問題文

これから以下の操作をちょうど N 回行います。

• K 個のロボットそれぞれについて、「進む」か「止まる」かを確率 \frac{1}{2} で決める。「進む」と決めたロボットたちは同時に正の方向に 1 進み、「止まる」と決めたロボットたちはその場から動かない。

ただし、すべての確率的な決定は独立であるとします。

### 制約

• 2 \leq K \leq 10
• 1 \leq N \leq 1000
• 0 \leq x_1 \lt x_2 \lt \cdots \lt x_K \leq 1000
• 入力は全て整数

### 入力

K N
x_1 x_2 \ldots x_K

2 2
1 2

### 出力例 1

374341633

374341633 \times 8 \equiv 5\pmod{998244353} ですので、374341633 を出力します。

2 2
10 100

1

### 入力例 3

10 832
73 160 221 340 447 574 720 742 782 970

553220346