Try this beautiful problem from American Invitational Mathematics Examination I, AIME I, 1999 based on Probability in games.

Probability in Games – AIME I, 1999 Question 13


Forty teams play a tournament in which every team plays every other team exactly once. No ties occur,and each team has a 50% chance of winning any game it plays.the probability that no two team win the same number of games is \(\frac{m}{n}\) where m and n are relatively prime positive integers, find \(log_{2}n\)

  • 10
  • 742
  • 30
  • 11

Key Concepts


Probability

Theory of equations

Combinations

Check the Answer


But try the problem first…

Answer: 742.

Source
Suggested Reading

AIME, 1999 Q13

Course in Probability Theory by Kai Lai Chung .

Try with Hints


First hint

\({40 \choose 2}\)=780 pairings with \(2^{780}\) outcomes

Second hint

no two teams win the same number of games=40! required probability =\(\frac{40!}{2^{780}}\)

Final Step

the number of powers of 2 in 40!=[\(\frac{40}{2}\)]+[\(\frac{40}{4}\)]+[\(\frac{40}{8}\)]+[\(\frac{40}{16}\)]+[\(\frac{40}{32}\)]=20+10+5+2+1=38 then 780-38=742.

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