Formula to find out total number of matches in league tournament :

Question

Accepted Answer

The correct option is (4):
The correct formula to calculate the total number of matches in a single league tournament (also known as a Round Robin tournament), where every team plays every other team once, is $\frac{N(N-1)}{2}$. This formula is derived from the fundamental principle of combinations in mathematics. If you have N teams, the total number of unique pairings (matches) is equivalent to selecting 2 teams out of N, which is represented by "N choose 2." The logic behind the numerator, $N(N-1)$, is that each of the N teams will play N-1 opponents. The division by 2 is essential because when you multiply $N \times (N-1)$, you are counting the match between Team A and Team B, and also the match between Team B and Team A, essentially counting every unique match twice. By dividing by 2, we arrive at the precise number of unique matches required for a complete single league fixture.

The incorrect option is (1):
The formula $\frac{N-1}{2}$ is conceptually flawed for calculating the total number of matches. In a league tournament with $N$ teams, the total number of matches is always much higher than a simple manipulation of $N-1$. For any tournament structure involving more than three teams, this formula results in a value significantly less than the actual number of matches, or, for an odd number of teams, it results in a non-integer value, which is impossible for the count of physical matches.

The incorrect option is (2):
The formula $\frac{N+1}{2}$ is also incorrect. It has no mathematical basis for calculating the total number of matches in a league tournament. As with Option 1, for an odd number of teams, this expression will result in a decimal value (e.g., for $N=5$, $6/2 = 3$), which cannot represent the total count of matches when the correct formula yields $\frac{5(4)}{2} = 10$. It is mathematically inconsistent with the combinatorial requirement of a round-robin format.

The incorrect option is (3):
The formula $\frac{N(N+1)}{2}$ is mathematically significant, as it calculates the sum of all integers from 1 to $N$ ($1+2+3+...+N$). However, it is not the correct formula for a single league tournament. In a single league, each of the $N$ teams plays $N-1$ matches, resulting in $N(N-1)/2$. This formula incorrectly calculates the number of matches for a single league tournament and is therefore unsuitable for this specific application in physical education tournament planning.