A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?
Problem
Source: 2012 Indonesia Round 2 TST 1 Problem 2
Tags: combinatorics proposed, combinatorics
chaotic_iak
26.02.2012 04:00
WARNING: Post contains spoilers Some people here argued about the wording. The last sentence, in Indonesian, read: Quote: Dalam berapa cara penilaian dapat diberikan kepada Candra, dengan asumsi bahwa apa yang dikatakan Candra adalah benar? For obvious reasons, I change Candra (name of person) to C. Translating directly: Quote: In how many ways can C gets his scores, with the assumption that C tells the truth? There are different interpretations:
$40452501600 = 3^8 \cdot \binom{8}{4} \cdot 2^4 \cdot 5505$
.
$5505 = \binom{4}{0} \cdot 2^4 \cdot 81 + \binom{4}{1} \cdot 2^3 \cdot 79 + \binom{4}{2} \cdot 2^2 \cdot 63 + \binom{4}{3} \cdot 2^1 \cdot 21 + \binom{4}{4} \cdot 2^0 \cdot 1$
.
$6561 = 3^8$
. Which of those is correct? Preferably, only respond if you have solved the problem in one of the interpretations...