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:
Example 1:
Input: n = 1, x = 2, y = 3
Output: 6
Explanation:
Example 2:
Input: n = 5, x = 2, y = 1
Output: 32
Explanation:
Example 3:
Input: n = 3, x = 3, y = 4
Output: 684
Constraints:
1 <= n, x, y <= 1000The key idea in #3317 Find the Number of Possible Ways for an Event is to model the counting process using combinatorics and dynamic programming. Instead of enumerating all possible configurations, which would be infeasible for large inputs, we compute the number of valid arrangements using mathematical relationships between selections and assignments.
A common strategy is to use dynamic programming where dp[i][j] represents the number of ways to organize the first i participants or elements into j groups or roles. Combinatorial tools such as binomial coefficients, permutations, and modular exponentiation help efficiently count arrangements. Precomputing factorials and combinations under a modulus significantly speeds up repeated calculations.
By carefully structuring transitions and combining them with combinatorial formulas, we avoid brute-force enumeration. The optimized solution typically runs in around O(n × k) time with O(n × k) space, depending on the DP formulation.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Dynamic Programming with Combinatorics | O(n × k) | O(n × k) |
| Optimized DP with Precomputed Combinations | O(n × k) | O(k) to O(n × k) |
NeetCode
Use these hints if you're stuck. Try solving on your own first.
Fix the number of stages.
Assign the Performers to the stages.
Use inclusion-exclusion to ensure that no stage has 0 performers.
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Jot down your thoughts, approach, and key learnings
Hard combinatorics and dynamic programming problems similar to this one frequently appear in FAANG-style interviews. They test mathematical reasoning, optimization, and the ability to model counting problems efficiently.
The optimal approach combines dynamic programming with combinatorial mathematics. Instead of enumerating every arrangement, we count valid configurations using DP states and precomputed combinations or factorial values under modular arithmetic.
Key concepts include combinatorics, dynamic programming, binomial coefficients, and modular arithmetic. Understanding how to count groupings and assignments efficiently is essential for building the DP transitions.
A 2D dynamic programming table is commonly used to store intermediate results. This structure helps track the number of ways to build valid configurations incrementally.