#2860

Medium

LeetCode Problem Workspace

Happy Students

The Happy Students problem asks how many ways a teacher can select a group of students so everyone is happy, based on certain conditions.

array sorting enumeration

Problem Statement

You are given a 0-indexed integer array nums representing students in a class. The array length n represents the total number of students, and each value nums[i] represents the happiness condition for the ith student.

A student becomes happy if either no student is selected or if all students with the same value are selected together. The task is to return the number of ways the teacher can select a group of students such that all students are happy.

Examples

Example 1

Input: nums = [1,1]

Output: 2

The two possible ways are: The class teacher selects no student. The class teacher selects both students to form the group. If the class teacher selects just one student to form a group then the both students will not be happy. Therefore, there are only two possible ways.

Example 2

Input: nums = [6,0,3,3,6,7,2,7]

Output: 3

The three possible ways are: The class teacher selects the student with index = 1 to form the group. The class teacher selects the students with index = 1, 2, 3, 6 to form the group. The class teacher selects all the students to form the group.

Constraints

1 <= nums.length <= 105
0 <= nums[i] < nums.length

Solution Approach

Group students by value

Start by grouping students by their happiness value in nums. This enables tracking all students with the same value, which must either be included or excluded together for the group to be valid.

Sort and count unique values

Sort the array to group consecutive students with the same happiness value. Then, count how many unique values exist, and for each value, calculate the number of possible selections of groups.

Evaluate possible groupings

For each unique student value, consider the number of ways to either include or exclude all students with that value in a group. Multiply the valid combinations to determine the total number of ways to form a happy student group.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach used to group and sort the students. Grouping the students takes linear time, while sorting the array takes O(n log n). The space complexity depends on the number of unique student values and how groups are stored during evaluation.

What Interviewers Usually Probe

Look for an understanding of how sorting and grouping can simplify the problem.
Check if the candidate is able to identify the key observation about grouping students with the same happiness value.
Assess whether the candidate can correctly evaluate the number of ways to form valid student groups.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly group students by their happiness value and considering invalid groupings.
Overcomplicating the problem by missing the key observation that students with the same value must either be included together or excluded.
Misunderstanding the time complexity and assuming the problem can be solved in linear time without sorting.

Follow-up variants

Change the problem to allow partial selections of students, introducing more complexity in the valid groupings.
Alter the input such that students have a wider range of happiness values, which increases the number of possible groupings.
Introduce constraints that limit the size of groups, requiring a more efficient solution to handle large inputs.

FAQ

What is the main pattern in the Happy Students problem?

The main pattern is array manipulation combined with sorting. Grouping students by their happiness values and sorting helps simplify the problem.

How do I handle large arrays in this problem?

By sorting the array first and grouping students by their happiness value, you can efficiently calculate the number of valid groupings, even for large arrays.

Can I solve this problem without sorting the array?

Sorting is crucial to group students with the same happiness value together. Without sorting, it would be much harder to evaluate valid groupings.

What makes this problem difficult?

The difficulty comes from identifying the correct grouping of students and understanding the requirement that students with the same happiness value must either be included or excluded together.

How does grouping students help in solving this problem?

Grouping students by their happiness value allows you to reduce the problem complexity and avoid unnecessary calculations by focusing on valid groupings of students with the same value.

terminal

Solution

Solution 1: Sorting + Enumeration

Assume that $k$ students are selected, then the following conditions hold:

1

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class Solution:
    def countWays(self, nums: List[int]) -> int:
        nums.sort()
        n = len(nums)
        ans = 0
        for i in range(n + 1):
            if i and nums[i - 1] >= i:
                continue
            if i < n and nums[i] <= i:
                continue
            ans += 1
        return ans

Continue Practicing

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