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How Many Ways Can the Party Win

Level: L3-L5 Topics: Dynamic Programming, Counting, Subset Sum Variants

Problem Statement

An election has N states, each with a known number of electoral votes. A party wins the election by receiving more than half of the total electoral votes. Count the number of distinct ways the party can win — that is, the number of subsets of states whose electoral votes sum to more than half the total.

Background & Constraints

  • Each state has a positive integer number of electoral votes.
  • The party wins if its total exceeds T/2 where T is the grand total of all electoral votes.
  • Two ways are distinct if the set of states won differs (even if the vote totals are the same).
  • N can be up to a few hundred.
  • Individual state vote counts can range from 1 to several hundred.

Examples

Example 1:

States: [3, 5, 2]
Total = 10, need > 5 to win.

All subsets and their sums:
{3} = 3, {5} = 5, {2} = 2
{3,5} = 8 > 5  -- WIN
{3,2} = 5, {5,2} = 7 > 5  -- WIN
{3,5,2} = 10 > 5  -- WIN

Number of ways to win: 3

Example 2:

States: [10, 8, 6, 4, 2]
Total = 30, need > 15 to win.

Some winning subsets: {10,8}, {10,6,4}, {10,8,6}, {10,8,6,4}, etc.
(Full enumeration omitted — use DP to count.)

Hints & Common Pitfalls

  • DP on subset sums: Build a DP table where dp[i][s] = the number of subsets using the first i states that sum to exactly s. Then the answer is the sum of dp[N][s] for all s > T/2.
  • Space optimization: You only need the previous row, so you can use a 1D DP array (iterate sums in reverse when updating).
  • Common mistake: Off-by-one in the winning threshold. "More than half" is strictly greater than T/2, not greater than or equal to. Be precise about whether ties count.
  • Common mistake: Using combinations formula instead of DP. The states have different vote counts, so you cannot use simple combinatorics.
  • The numbers can get very large — consider whether you need big integer arithmetic or modular arithmetic.

Follow-Up Questions

  1. What if ties also count as wins? How does the count change?
  2. Count the number of ways the opponent wins instead. What's the relationship between the two counts?
  3. Generalize the input: instead of a fixed list of states, what if each state can have a variable number of votes within a range?
  4. What is the time complexity of your solution? What is the space complexity?
  5. Can you use a meet-in-the-middle approach to handle larger N (e.g., N up to 40) where the sum range is too large for standard DP?