201135: [AtCoder]ARC113 F - Social Distance
Description
Score : $1000$ points
Problem Statement
Given is an integer sequence of length $N+1$: $X_0,X_1,\ldots,X_N$, where $0=X_0 < X_1 < \ldots < X_N$ holds.
Now, $N$ people numbered $1$ through $N$ will appear on a number line. Person $i$ will appear at a real coordinate chosen uniformly at random from the interval $[X_{i-1},X_i]$.
Find the expected value of the smallest distance between two people, modulo $998244353$.
Definition of the expected value modulo $998244353$
We can prove that the expected value in question is always a rational number. We can also prove that, under the constraints of this problem, if we express the expected value as an irreducible fraction $\frac{P}{Q}$, we have $Q \neq 0 \pmod{998244353}$. Thus, there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Report this $R$.
Constraints
- $2 \leq N \leq 20$
- $0=X_0 < X_1 < \cdots < X_N \leq 10^6$
Input
Input is given from Standard Input in the following format:
$N$ $X_0$ $X_1$ $\ldots$ $X_N$
Output
Print the expected value of the smallest distance between two people, modulo $998244353$.
Sample Input 1
2 0 1 3
Sample Output 1
499122178
There are just two people, so the expected value of the smallest distance between two people is just the expected value of the distance between Person $1$ and Person $2$. The answer is $3/2$.
Sample Input 2
5 0 3 4 8 9 14
Sample Output 2
324469854
The answer is $196249/172800$.
Sample Input 3
20 0 38927 83112 125409 165053 204085 246405 285073 325658 364254 406395 446145 485206 525532 563762 605769 644863 683453 722061 760345 798556
Sample Output 3
29493181