 ## 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.

## 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.