Total number of Byes will be in First round of 33 teams of single knock out tournament:

Question

Accepted Answer

The correct option is (3):

The correct answer is 31 because the number of Byes in a single knock-out tournament is calculated by subtracting the total number of participating teams from the next immediate power of two. A single knock-out tournament is fundamentally structured to operate with teams in multiples of 2, such as 4, 8, 16, 32, 64, and so on. Since the number of teams, $N$, is 33, which is not a power of two, we must determine the next highest power of two, which in this case is $2^6 = 64$. The calculation is performed using the formula: Total Byes = (Next Power of Two) - N. Substituting the values, we get $64 - 33 = 31$ Byes. These 31 Byes allow 31 teams to advance directly to the second round without playing a match in the first round, thus reducing the number of competing teams to 32 (a power of two) for the second round onwards.

The incorrect option is (1):
The value 21 is mathematically irrelevant to the standard knock-out fixture drawing for 33 teams. The number of Byes is strictly determined by the difference between the total teams and the next power of two, a principle necessary to maintain a balanced bracket and the single-elimination structure. Choosing 21 Byes would leave $33 - 21 = 12$ teams to play 6 matches, resulting in 6 winners. When these 6 winners are added to the 21 teams who received a Bye, the second round would have $21 + 6 = 27$ teams. Since 27 is not a power of two, this calculation fails to prepare the tournament for the subsequent rounds in the required elimination format.

The incorrect option is (2):
The number 1 is far too small to balance the tournament bracket. If only 1 Bye were given, the remaining $33 - 1 = 32$ teams would be available to play matches. These 32 teams would play 16 matches, resulting in 16 winners. When the 1 winner from the Bye is added, the second round would have $16 + 1 = 17$ teams. Since 17 is not a power of two, the tournament structure would become immediately unbalanced in the second round, requiring further ad-hoc calculations for Byes, which contradicts the goal of fixing all Byes in the first round.

The incorrect option is (4):
While 32 is a significant power of two ($2^5$), it is not the correct number of Byes because it is less than the total number of teams (33). The calculation for Byes must target the *next highest* power of two, which is 64. Giving 32 Byes would only leave $33 - 32 = 1$ team to play in the first round, which is logically incorrect for setting up the elimination bracket. Furthermore, the goal is to reach a total pool of 64 slots, not 32, so the required number of Byes must be calculated using 64 as the base, leading back to $64 - 33 = 31$.