Proving same score given sum of score using PHP

Question

Encountered this question while practicing. Sadly there was no solution given. Sounds like a PHP question but with so little information given I am wondering how it can be solved. Any hints/help would be greatly appreciated. Thanks.

21 students took an exam and their scores sum to 200. If the scores are nonnegative integers, prove that there are two students with the same score.

If they all have different scores, what is the lowest possible sum? — Rushy, Apr 30 '21 at 18:47
If all different score lowest sum will be 0+1+2+...+20 = 210 so it contradicts. Is that the way? — ho03lesin, Apr 30 '21 at 18:52
Pigeonhole principle is mostly a discrete form of the law of averages. Either all data points are equal, or to get the average you need 1 above the average and at least 1 below the average. — Roddy MacPhee, Jun 10 '21 at 23:00

user2005142 · Answer 1 · 2021-04-30T18:58:08.120

0

This can be shown by a contradiction. Assume for the contradiction that no two students received the same same score. Then, a lower bound on the total sum of scores is \begin{align} \sum_{i=1}^{21} = \frac{1}{2}21\cdot 22 = 231 > 200. \end{align}

Hence at least two students received the same score.

(Or, assuming scores include 0, the lower bound on the total score is $\frac{1}{2}20\cdot 21 = 210.) $

edited Apr 30 '21 at 18:58

answered Apr 30 '21 at 18:52

user2005142

91

1

Thanks! Appreciate it – ho03lesin Apr 30 '21 at 18:56

Proving same score given sum of score using PHP

1 Answers1