Newcastle: Mathematical Hopes for Champions League and Relegation

With three games left in the Premier League, Newcastle finds itself in a paradoxical situation: it has mathematical hopes for both Champions League qualification and relegation.
This year's Premier League is extremely exciting. Arsenal and Manchester City are battling for the title, while teams like Tottenham, West Ham, Nottingham Forest, and Leeds are trying to avoid relegation. There is also a scramble for European spots.
Newcastle needs three wins in its remaining games, combined with specific results from Bournemouth, Brentford, and Brighton, to finish in sixth place. Additionally, Aston Villa must finish fifth and win the Europa League for sixth place to grant an extra ticket to the Champions League.
For Newcastle to be relegated, Tottenham and West Ham must achieve three wins, Leeds, Crystal Palace and Nottingham must gather 3-4 points, and Newcastle must lose all of its remaining games.