Contest Score Calculator

Algebra Level 3

A contest score calculator is programmed to mark students' contest. The contest has 25 questions. Correct answers are worth 6 points. Omitted questions are worth 2 points. Incorrect answers worth 0 points. How many different scores are possible in this contest?


The answer is 75.

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2 solutions

Chew-Seong Cheong
Oct 21, 2019

Let the number of questions omitted be m m and the number of questions answered correctly be n n . Then the score is given by s ( m , n ) = 2 m + 6 n s(m,n) = 2m + 6n , where 0 m 25 0 \le m \le 25 and n 25 m n \le 25-m . We can table s ( m , n ) s(m,n) as follows:

m = 0 1 2 3 22 23 24 25 n = 0 0 2 4 6 44 46 48 50 1 6 8 10 12 50 52 54 2 12 14 16 18 56 58 3 18 20 22 24 62 22 132 134 136 138 23 138 140 142 24 144 146 25 150 \begin{array} {rrrrrrrrr} | & m = 0 & 1 & 2 & 3 & \cdots & 22 & 23 & 24 & 25 \\ \hline n = 0 | & \blue 0 & \blue 2 & \blue 4 & \blue 6 & \cdots & \blue{44} & \blue{46} & \blue{48} & \blue{50} \\ 1 | & 6 & 8 & 10 & 12 & \cdots & 50 & \blue{52} & \blue{54} \\ 2 | & 12 & 14 & 16 & 18 & \cdots & \blue{56} & \blue{58} \\ 3 | & 18 & 20 & 22 & 24 & \cdots & \blue{62} \\ \cdots | & \cdots & \cdots & \cdots & \cdots & \cdots \\ 22 | & 132 & 134 & \blue{136} & \blue{138} \\ 23 | & 138 & \blue{140} & \blue{142} \\ 24 | & \blue{144} & \blue{146} \\ 25 | & \blue{150} \end{array}

From the table, it can be seen that the score s ( m . n ) s(m.n) takes all even numbers from 0 to 150 except 148. Therefore there are 75 \boxed{75} different scores.

Kevin Xu
Oct 20, 2019

The score could be categorized into three types: \\ 6 k 0 k 25 6k \quad 0 \leq k \leq 25\\ 6 k + 2 0 k 24 6k+2 \quad 0 \leq k \leq 24\\ 6 k + 4 0 k 23 6k+4 \quad 0 \leq k \leq 23\\

Why? \\ Answers that are either correct or wrong has a score of multiple of 6 \\ Answers that are either correct or wrong or have 3 n 3n numbers of omitted questions has a score of multiple of 6 \\ Answers that are either correct or wrong or have 3 n + 1 3n+1 numbers of omitted questions has a score of (multiple of 6 ) + 2 \\ Answers that are either correct or wrong or have 3 n + 2 3n+2 numbers of omitted questions has a score of (multiple of 6 ) + 4 \\

That there could be no other outcomes \\ So we sum the outcomes up: 26 + 25 + 24 = 75 26 + 25 + 24 = 75

Nice solution (and problem). You can get any even score from 0 0 to 150 150 inclusive except 148 148 .

Chris Lewis - 1 year, 7 months ago

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