3317. Find the Number of Possible Ways for an Event
Description
You are given three integers n
, x
, and y
.
An event is being held for n
performers. When a performer arrives, they are assigned to one of the x
stages. All performers assigned to the same stage will perform together as a band, though some stages might remain empty.
After all performances are completed, the jury will award each band a score in the range [1, y]
.
Return the total number of possible ways the event can take place.
Since the answer may be very large, return it modulo 109 + 7
.
Note that two events are considered to have been held differently if either of the following conditions is satisfied:
- Any performer is assigned a different stage.
- Any band is awarded a different score.
Example 1:
Input: n = 1, x = 2, y = 3
Output: 6
Explanation:
- There are 2 ways to assign a stage to the performer.
- The jury can award a score of either 1, 2, or 3 to the only band.
Example 2:
Input: n = 5, x = 2, y = 1
Output: 32
Explanation:
- Each performer will be assigned either stage 1 or stage 2.
- All bands will be awarded a score of 1.
Example 3:
Input: n = 3, x = 3, y = 4
Output: 684
Constraints:
1 <= n, x, y <= 1000
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ to represent the number of ways to arrange the first $i$ performers into $j$ programs. Initially, $f[0][0] = 1$, and the rest $f[i][j] = 0$.
For $f[i][j]$, where $1 \leq i \leq n$ and $1 \leq j \leq x$, we consider the $i$-th performer:
- If the performer is assigned to a program that already has performers, there are $j$ choices, i.e., $f[i - 1][j] \times j$;
- If the performer is assigned to a program that has no performers, there are $x - (j - 1)$ choices, i.e., $f[i - 1][j - 1] \times (x - (j - 1))$.
So the state transition equation is:
$$ f[i][j] = f[i - 1][j] \times j + f[i - 1][j - 1] \times (x - (j - 1)) $$
For each $j$, there are $y^j$ choices, so the final answer is:
$$ \sum_{j = 1}^{x} f[n][j] \times y^j $$
Note that since the answer can be very large, we need to take the modulo $10^9 + 7$.
The time complexity is $O(n \times x)$, and the space complexity is $O(n \times x)$. Here, $n$ and $x$ represent the number of performers and the number of programs, respectively.
1 2 3 4 5 6 7 8 9 10 11 12 13 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
|