Ex - A Nameless Counting Problem

AtCoder

IDabc288_h

Time3000ms

Memory256MB

Difficulty—

Print the number of integer sequences of length $N$, $A = (A_1, A_2, \ldots, A_N)$, that satisfy both of the following conditions, modulo $998244353$. * $0 \leq A_1 \leq A_2 \leq \cdots \leq A_N \leq M$ * $A_1 \oplus A_2 \oplus \cdots \oplus A_N = X$ Here, $\oplus$ denotes bitwise XOR. What is bitwise XOR?The bitwise XOR of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows. * When $A \oplus B$ is written in binary, the $k$\-th lowest bit ($k \geq 0$) is $1$ if exactly one of the $k$\-th lowest bits of $A$ and $B$ is $1$, and $0$ otherwise. For instance, $3 \oplus 5 = 6$ (in binary: $011 \oplus 101 = 110$). ## Constraints * $1 \leq N \leq 200$ * $0 \leq M \lt 2^{30}$ * $0 \leq X \lt 2^{30}$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $X$ [samples]

Samples

Input #1

3 3 2

Output #1

5

Here are the five sequences of length $N$ that satisfy both conditions in the statement: $(0, 0, 2), (0, 1, 3), (1, 1, 2), (2, 2, 2), (2, 3, 3)$.

Input #2

200 900606388 317329110

Output #2

788002104

API Response (JSON)

{
  "problem": {
    "name": "Ex - A Nameless Counting Problem",
    "description": {
      "content": "Print the number of integer sequences of length $N$, $A = (A_1, A_2, \\ldots, A_N)$, that satisfy both of the following conditions, modulo $998244353$. *   $0 \\leq A_1 \\leq A_2 \\leq \\cdots \\leq A_N \\l",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc288_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Print the number of integer sequences of length $N$, $A = (A_1, A_2, \\ldots, A_N)$, that satisfy both of the following conditions, modulo $998244353$.\n\n*   $0 \\leq A_1 \\leq A_2 \\leq \\cdots \\leq A_N \\l...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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