Question

In a certain game, each of 5 players received a score between 0 and 100, inclusive. If their average (arithmetic mean) score was 80, what is the greatest possible number of the 5 players who could have received a score of 50?
(a) None
(b) One
(c) Two
(d) Three

Answer 1

Hint: Assume \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\] and \[{{x}_{5}}\] as the score of the 5 players respectively. Consider different cases in which assume that ‘n’ players obtained a score of 50, where n = 0, 1, 2, 3, 4, 5. Find the total score of the five players by using the formula: - total score = mean \[\times \] number of players. Find the values of ‘n’ for which the assumed cases are possible and hence find the greatest value of ‘n’ to get the answer.

Complete step by step solution:
Here, we have been provided with the average score of 5 players in a certain game. We have been asked to determine the greatest possible number of 5 players who could have received a score of 50.
Now, let us assume \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\] and \[{{x}_{5}}\] as the scores of 5 players respectively. It is said that they scored between 0 and 100, including 0 and 100 also. Since, the average of these five scores is 80, we have,
\[\Rightarrow \] Mean = (total score / number of players)
\[\Rightarrow \] Total score = mean \[\times \] number of players
\[\Rightarrow \] Total score = 80 \[\times \] 5
\[\Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}=400\]
Now, let us consider that ‘n’ number of players scored 50, where ‘n’ can be 0, 1, 2, 3, 4 or 5 that we need to check. So, considering different cases that may arise, we have,
(i) n = 0
When none of the players scored 50 that means they can fulfil the condition: - \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}=400\]. For example, if each player will score 80 then we will get the given condition satisfied. So, the value n = 0 is correct.
(ii) n = 1
When 1 player scored 50 then we have the condition: -
\[\Rightarrow 50+{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=400\]
\[\Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=350\]
So, the remaining four players have to score 350 which is possible because they can score a maximum of 100 that means they can sum the score up to 400. So, the value n = 1 is correct.
(iii) n = 2
When 2 players scored 50 then we have the condition: -
\[\begin{align}
  & \Rightarrow 50+50+{{x}_{1}}+{{x}_{2}}+{{x}_{3}}=400 \\
 & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=300 \\
\end{align}\]
So, the remaining three players have to score 300 which is possible if all these remaining players will score 100 each. So, the value n = 2 is correct.
(iv) n = 3
When 3 players scored 50 then we have the condition: -
\[\begin{align}
  & \Rightarrow 50+50+50+{{x}_{1}}+{{x}_{2}}=400 \\
 & \Rightarrow {{x}_{1}}+{{x}_{2}}=250 \\
\end{align}\]
This condition is not possible because even if the two players will score 100 each, they will sum up to 200. So, n = 3 is incorrect.
(v) n = 4
When four players scored 50 then we have the condition: -
\[\begin{align}
  & \Rightarrow 50+50+50+50+{{x}_{1}}=400 \\
 & \Rightarrow {{x}_{1}}=200 \\
\end{align}\]
The above condition is not possible because \[0\le {{x}_{1}}\le 100\]. So, the value n = 4 is incorrect.
(vi) n = 5
When 5 players scored 50 then we have the condition: -
\[\begin{align}
  & \Rightarrow 50+50+50+50+50=400 \\
 & \Rightarrow 250=400 \\
\end{align}\]
The above condition cannot be possible, so the value n = 5 is incorrect.
On observing all the conditions and different cases we can conclude that the possible values of ’n’ are: - n = 0, 1 and 2. Clearly, we can see that the greatest value of ‘n’ is 2.
Hence, option (c) is the correct answer.

Note: One may note that we cannot take the value of n greater than 5 because there are only 5 players. You may note that option (a) and (b), i.e., the values n = 0 and n = 1, are also correct because they satisfy the given conditions but since we have been asked to determine the maximum number of players possible that is why we have considered only n = 2 as our answer.