AMC 10 USA Math Olympiad

Probability Problem from AMC 10A – 2020 – Problem No. 15

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

What is Probability?

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.

Try the Problem from AMC 10 – 2020

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.

Knowledge Graph

Probability- knowledge graph

Use some hints

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.

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