Ex - Random Painting

AtCoder

IDabc242_h

Time3000ms

Memory256MB

Difficulty—

There are $N$ squares numbered $1$ to $N$. Initially, all squares are painted white. Additionally, there are $M$ balls numbered $1$ to $M$ in a box. We repeat the procedure below until all squares are painted black. 1. Pick up a ball from a box uniformly at random. 2. Let $x$ be the index of the ball. Paint Squares $L_x, L_x+1, \ldots, R_x$ black. 3. Return the ball to the box. Find the expected value of the number of times the procedure is done, modulo $998244353$ (see Notes). ## Constraints * $1 \leq N,M \leq 400$ * $1 \leq L_i \leq R_i \leq N$ * For every square $i$, there is an integer $j$ such that $L_j \leq i \leq R_j$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $L_1$ $R_1$ $L_2$ $R_2$ $\hspace{0.5cm}\vdots$ $L_M$ $R_M$ [samples] ## Notes It can be proved that the sought expected value is always a rational number. Additionally, under the Constraints of 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 $0 \leq R \lt 998244353$. You should find this $R$.

Samples

Input #1

Output #1

499122180

The sought expected value is $\frac{7}{2}$.
We have $499122180 \times 2 \equiv 7\pmod{998244353}$, so $499122180$ should be printed.

Input #2

Output #2

Input #3

Output #3

499122193

API Response (JSON)

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      "content": "There are $N$ squares numbered $1$ to $N$. Initially, all squares are painted white.\nAdditionally, there are $M$ balls numbered $1$ to $M$ in a box.\nWe repeat the procedure below until all squares are...",
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