Question

In a tournament with five teams, each team plays against every other team exactly once. Each game is won by one of the playing teams and the winning team scores one point, while the losing team scores zero, which of the following is not necessarily true.
A) There are at least two teams which have at most two points each.
B) There are at least two teams which have at least two points each.
C) There are at most three teams which have at least three points each.
D) There are at most four teams which have at most two points each.

Answer 1

Hint: In this type of problem we simply look into given options and solve the problem by pointing out every option, denote all the five teams by any variable and make a table for all the five teams for losing and winning. That table helps to approach the required result.

Complete step-by-step answer:
We have given the total numbers of teams which are 5 and each team plays against every other team exactly once and the winning team scores one point and the losing team scores zero.
The goal is to verify that if the given statement is true about our question.
First of all, we will make a table in which each team can score at least two points.
Let, five teams are${T_1}$ , ${T_2}$, ${T_3}$, ${T_4}$, and ${T_5}$.
The winning point is 1 and the losing point is zero, then the table given below shows that every team can win two matches. As there are 5 teams and each team has four choices to play as given that each team plays against every other team exactly once. Assuming that each team won two games and lost two games.
Table is shown below:

Match number	${T_1}$	${T_2}$	${T_3}$	${T_4}$	${T_5}$
1	1	0
2	1		0
3	0			1
4	0				1
5		1	0
6		1		0
7		0			1
8			1	0
9			1		0
10				1	0
total	2	2	2	2	2

For, option (A), (B), (C) are satisfying because according to the table, each team can score at least two points.
It contradicts option (D).
In option (D), it says that there at most four teams which have at most two points each.
If four teams score 2 points it means the 5th team score is also 2.
So, option (D) contradicts.
So, option (D) is the required answer.

Note: Make the table carefully, we made the table of at least two winning matches that are followed according to the requirements of the options. There are two words used in the given options, “at least and at most”. In case of at least, the minimum requirement must be satisfied and in case of at most, the maximum requirement must be satisfied.