What is the optimal time complexity of solving Count ways to reach score?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Count ways to reach score?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Count ways to reach score?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Count ways to reach score?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingEasy

Count ways to reach score

Learn how to solve the 'Count ways to reach score' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function count_ways_score(n) that returns the number of distinct combinations of moves to reach a score n in a game where a player can score 3, 5, or 10 points in each move. Note that combinations with different ordering of moves are considered the same (e.g., scoring 3 then 5 is the same combination as scoring 5 then 3).

Constraints

•1 <= n <= 1000

Examples

Example 1

Input

count_ways_score(13)

Output

Explanation

There are 2 combinations to reach 13: {3, 5, 5} and {3, 10}.

Example 2

Input

count_ways_score(20)

Output

Explanation

There are 4 combinations to reach 20: {10, 10}, {5, 5, 10}, {5, 5, 5, 5}, and {3, 3, 3, 3, 3, 5}.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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