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.
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