Q.Read the following passage and answer the questions that follow. The Physical Education teacher of Sunrise Public School has to organise the annual inter-house basketball tournament. Fifteen

(i) Total number of matches

For a knock-out tournament, the total number of matches is always one less than the number of teams, because each match eliminates exactly one team and only one champion remains.

Matches = N - 1 = 15 - 1 = 14

(ii) Number of byes

Byes are given when the number of teams is not a power of 2. The next higher power of 2 above 15 is 16.

Byes = (next power of 2) - N = 16 - 15 = 1

Only one bye is needed.

(iii) Upper half and lower half teams

Since N = 15 is odd, the upper half gets one more team than the lower half.

Upper half = (N + 1)/2 = (15 + 1)/2 = 8 teams

Lower half = (N - 1)/2 = (15 - 1)/2 = 7 teams

(iv) Procedure used to keep the two strongest teams apart …

1. Total number of matches In a single-elimination (knock-out) tournament, every team except the champion loses exactly once. That means the number of matches is always one fewer than the number of teams — because each match eliminates one team, and only one team remains undefeated at the end.

   Total matches = N - 1

   Here N = 15, so: Total matches = 15 - 1 = 14
2. Number of byes A bye is a free pass to the next round given to a team when the total number of teams is not a power of 2. The idea is to reduce the field to a power of 2 so that every round from then on has an even number of teams and no further byes are needed.

   Number of byes = (next higher power of 2) - N

   The powers of 2 are: 2, 4, 8, 16, 32, … For N = 15, the next higher power is 16. Byes = 16 - 15 = 1 So exactly one team gets a bye in the first round.
3. Teams in the upper half and lower half When the number of teams is odd, the halves cannot be equal. The standard rule is to put the extra team in the upper half.

   Upper half = (N + 1)/2, Lower half = (N - 1)/2 (when N is odd)

   For N = 15: Upper half = (15 + 1)/2 = 8 teams Lower half = (15 - 1)/2 = 7 teams
4. Procedure to keep Alpha House and Omega House apart until the final …