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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 **\(\frac {m}{n}\),

*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.

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. *

- https://www.cheenta.com/surface-area-of-a-cube-amc-10a-2020/
- https://www.youtube.com/watch?v=6Cn2P6SoIi0

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