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

Source
Competency
Difficulty
Suggested Book

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


First hint

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.

Second Hint

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.

Final Step

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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