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?
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The score could be categorized into three types: 6 k 0 ≤ k ≤ 2 5 6 k + 2 0 ≤ k ≤ 2 4 6 k + 4 0 ≤ k ≤ 2 3
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 numbers of omitted questions has a score of multiple of 6 Answers that are either correct or wrong or have 3 n + 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 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: 2 6 + 2 5 + 2 4 = 7 5
Nice solution (and problem). You can get any even score from 0 to 1 5 0 inclusive except 1 4 8 .
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Let the number of questions omitted be m and the number of questions answered correctly be n . Then the score is given by s ( m , n ) = 2 m + 6 n , where 0 ≤ m ≤ 2 5 and n ≤ 2 5 − m . We can table s ( m , n ) as follows:
∣ n = 0 ∣ 1 ∣ 2 ∣ 3 ∣ ⋯ ∣ 2 2 ∣ 2 3 ∣ 2 4 ∣ 2 5 ∣ m = 0 0 6 1 2 1 8 ⋯ 1 3 2 1 3 8 1 4 4 1 5 0 1 2 8 1 4 2 0 ⋯ 1 3 4 1 4 0 1 4 6 2 4 1 0 1 6 2 2 ⋯ 1 3 6 1 4 2 3 6 1 2 1 8 2 4 ⋯ 1 3 8 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 2 2 4 4 5 0 5 6 6 2 2 3 4 6 5 2 5 8 2 4 4 8 5 4 2 5 5 0
From the table, it can be seen that the score s ( m . n ) takes all even numbers from 0 to 150 except 148. Therefore there are 7 5 different scores.