The Probability theory, a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
A positive integer divisor of 12! is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as \(\frac {m}{n}\), where m and n are relatively prime positive integers. What is m+n ?
A)3 B) 5 C)12 D) 18 E) 23
American Mathematics Competition 10 (AMC 10), {2020}, {Problem number 15}
Inequality (AM-GM)
6 out of 10
Secrets in Inequalities.
If you really need any hint try this out:
The prime factorization of 12! is \(2^{10}\cdot 3^{5}\cdot 5^{2}\cdot 7\cdot 11\)
This yields a total of \( 11\cdot 6 \cdot 3 \cdot 2 \cdot 2 \) divisors of 12!.
In order to produce a perfect square divisor, there must be an even exponent for each number in the prime factorization.
Again 7 and 11 can not be in the prime factorization of a perfect square because there is only one of each in 12!. Thus, there are \(6 \cdot 3\cdot 2\) perfect squares.
I think you already got the answer but if you have any doubt use the last hint :
So the probability that the divisor chosen is a perfect square is \(\frac {6.3 . 2}{11 . 6. 3. 2. 2} = \frac {1}{22}\)
\(\frac {m}{n} = \frac {1}{22} \)
m+n = 1+22 = 23.
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]