Скачать или смотреть Unfriendly: Combination Question with Constraints in Python | QuantGuide IO

Solve the "Unfriendly" problem from QuantGuide IO with this step-by-step combinatorics tutorial! The problem: John has 8 friends and wants to invite 5 to his party, but two of his friends have a conflict and refuse to attend together. We’ll calculate the number of possible guest combinations, which turns out to be 50. We’ll break down the problem, use combinations to find the total possibilities, subtract the unfavorable cases, and verify with Python. Perfect for math enthusiasts, quant interview prep, or anyone learning combinatorics with constraints!

🔍 *What You'll Learn:*
Understanding the Unfriendly problem setup
Calculating total combinations without constraints
Subtracting unfavorable combinations (when both conflicting friends are invited)
Computing the final result with Python

💻 *Solution Walkthrough:*
Total friends: 8, inviting 5.
Total ways to choose 5 friends without constraints: \( C(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4}{5 \times 4 \times 3 \times 2 \times 1} = 56 \).
Unfavorable cases (both conflicting friends are invited):
If the two conflicting friends are both invited, we must choose 3 more friends from the remaining 6 friends (8 - 2 = 6).
Ways to choose 3 friends from 6: \( C(6, 3) = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \).
Favorable cases = Total ways - Unfavorable ways = \( 56 - 20 = 36 \).
Final answer: 36 combinations.
Note: The code provided calculates \( C(8, 5) - C(6, 3) = 56 - 20 = 36 \), but let’s verify our understanding.

Python code to compute the combinations
import math

Total ways to choose 5 friends from 8
all_combinations = math.comb(8, 5) # C(8, 5) = 56

Unfavorable ways: both conflicting friends are invited, choose 3 more from the remaining 6
unfavorable = math.comb(6, 3) # C(6, 3) = 20

Favorable ways
favorable = all_combinations - unfavorable # 56 - 20 = 36
print(f"Total combinations: {all_combinations}")
print(f"Unfavorable combinations: {unfavorable}")
print(f"Favorable combinations: {favorable}")
Output:
Total combinations: 56
Unfavorable combinations: 20
Favorable combinations: 36

Alternative approach: Choose cases where at least one conflicting friend is not invited
- Case 1: Neither of the conflicting friends is invited: Choose 5 from the remaining 6: C(6, 5) = 6
- Case 2: Exactly one of the conflicting friends is invited: Choose 1 from the 2 conflicting friends, and 4 from the remaining 6: 2 * C(6, 4) = 2 * 15 = 30
Total = 6 + 30 = 36, which matches!

🌟 *Why Solve Unfriendly?*
This problem is a classic combinatorics question with constraints, perfect for quant finance interviews or math competitions! It teaches you how to handle exclusion constraints using combinations, a key skill for problem-solving. The number of possible guest combinations is 36, and we’ll show you two ways to get there. Master this, and you’ll be ready for more advanced combinatorics challenges!

📚 *Who’s This For?*
Math and combinatorics enthusiasts
Students prepping for quant interviews
Anyone practicing for math competitions

👍 Like, subscribe, and comment: What combinatorics problem should we solve next? Next up: More quant interview questions—stay tuned!

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