A multiple choice contest
Probability
Level
1
A math contest is made up of 52 multiple choice questions each worth either 0 (if wrong or no answer) or 1 (if right). How many students must write the test to be sure that at least 39 have the same final score?
The answer is 2015.
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Final score can be any integer between 0 and 52, which gives us 5 2 − 0 + 1 = 5 3 different possibilities. To guarantee that 39 students get the same score, there must be 38 students for each distinct score so that no matter what the last student scores, he will always end up being the 39th one. So basically this is only a basic application of the pigeonhole principle now. So we obtain the answer as 3 8 ∗ 5 3 + 1 = 2 0 1 5