Question

a positive integer divisor of 12! is chosen at random. the probability that the di-visor chosen is a perfect square can be expressed asmn, wheremandnare relativelyprime positive integers. what ism n?

Answer 1

The probability that the divisor chosen is a perfect square is found to be 1/22 which is m:n.

What exactly is meant by probability?

• Probability is synonymous with possibility. It is a mathematical branch that deals with the occurrence of a random event.
• The value ranges from zero to one. Probability has been introduced in mathematics to predict the likelihood of events occurring.

Given: Randomly, a positive integer divisor of 12! is selected.

Prime factorization of 12! = 2¹⁰ × 3⁵ × 5² × 7¹ × 11

Total divisors of 12! = 11 × 6 × 3 × 2 × 2

• Each number in the prime factorization must have an even exponent in order to construct a perfect square divisor.
• In the prime factorization, the divisor cannot have any factors of 7 and 11 because there is only one of each in 12!.
• There are 6 × 3 × 2 perfect squares as a result.

The probability that the divisor chosen is a perfect square is given as:

(6 × 3 × 2)/(11 × 6 × 3 × 2 × 2)

= 1/22

=m/n

Therefore, the probability that the divisor chosen is a perfect square is found to be 1/22 which is m:n.

Learn more about probability here:

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