Question

HOW MANY BYES WILL BE GIVEN IF 19 TEAMS ARE PARTICIPATING IN A KNOCK-OUT TOURNAMENT? *

17

16

14

13​

Answer 1

14 byes will be given

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO DETERMINE

The number of Byes Will be given if 19 teams are participating in a KNOCK-OUT tournament

CONCEPT TO BE IMPLEMENTED

In a tournament Bye is a privilege given to a team participating in the tournament which is performed generally by seeding it or by draw of lots

If the number of competing teams in the tournament are not a multiple of 2 then some teams may get BYE

PROCESS

Suppose there are N odd number of teams.

$\sf{Number \: of \: byes \: = {2}^{m} - N }$

$\sf{where \: m \: is \: the \: least \: integer \: satisfying \: \: \: {2}^{m} > N }$

CALCULATION

In a tournament a match is played between two teams

Now

$\sf{16 < 19 < 32 \: \: \: so \: that \: \: \: {2}^{4} < 19 < {2}^{5} \: }$

$\sf{ \therefore \: \: 5 \: \: is \: the \: least \: integer \: such \: that\: \: \: {2}^{5} > 19 }$

$\sf{Hence \: the \: number \: of \: Byes }$

$\sf{ = {2}^{5} - 19 \: }$

$= \sf{32 - 19}$

$= \sf{13}$

RESULT

The number of Byes will be given if 19 teams are participating in a KNOCK-OUT tournament = 13

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