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Three football teams are taking part in a tournament.
Each team plays each other team once.
For a win the team scores 3 points, the other team 0 points.
For a draw both teams get 1 point each.
Which number of points is impossible, for any team to reach at the end of this tournament?
(A) 1
(B) 2
(C) 4
(D) 5
(E) 6

Three football teams are taking part in a tournament. $\newline$ Each team plays each other team once. $\newline$ For a win the team scores $3$ points, the other team $0$ points. $\newline$ For a draw both teams get $1$ point each. $\newline$ Which number of points is impossible, for any team to reach at the end of this tournament? $\newline$ (A) $1$ $\newline$ (B) $2$ $\newline$ (C) $4$ $\newline$ (D) $5$ $\newline$ (E) $6$

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Compound events: find the number of sums

Full solution

Q. Three football teams are taking part in a tournament.

\newline

Each team plays each other team once.

\newline

For a win the team scores

3

points, the other team

0

points.

\newline

For a draw both teams get

1

point each.

\newline

Which number of points is impossible, for any team to reach at the end of this tournament?

\newline

(A)

1

\newline

(B)

2

\newline

(C)

4

\newline

(D)

5

\newline

(E)

6

Analyze Possible Outcomes: Let's analyze the possible outcomes for each match and the points that can be earned. $\newline$ - If a team wins both of its matches, it will earn $3$ points per match, totaling $6$ points. $\newline$ - If a team wins one match and draws the other, it will earn $3$ points for the win and $1$ point for the draw, totaling $4$ points. $\newline$ - If a team draws both matches, it will earn $1$ point per match, totaling $2$ points. $\newline$ - If a team loses one match and draws the other, it will earn $0$ points for the loss and $1$ point for the draw, totaling $1$ point. $\newline$ - If a team loses both matches, it will earn $0$ points.
Check Answer Options: Now let's check each answer option to see if it's possible for a team to reach that number of points: $\newline$ (A) $1$ point is possible by losing one match and drawing one match. $\newline$ (B) $2$ points are possible by drawing both matches. $\newline$ (C) $4$ points are possible by winning one match and drawing one match. $\newline$ (D) $5$ points are not listed in our possible outcomes, so we need to check if it's possible to achieve this score. $\newline$ (E) $6$ points are possible by winning both matches.
Check Possibility of $5$ Points: To check if $5$ points are possible, we need to consider the combinations of wins, draws, and losses that could lead to this total: $\newline$ - Winning both matches gives $6$ points, not $5$ . $\newline$ - Winning one match ( $3$ points) and drawing one match ( $1$ point) gives $4$ points, not $5$ . $\newline$ - Drawing both matches gives $2$ points, not $5$ . $\newline$ - It is not possible to reach exactly $5$ points with any combination of wins, draws, and losses in two matches.

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