Question

Two numbers are selected from the set of integers 1 to 25 . what is the probability that the product of the numbers will be 36 . (1) \(\frac{1}{200}\) (2) \(\frac{1}{100}\) (3) \(\frac{1}{50}\) (4) \(\frac{1}{75}\)

Short answer

Answer: \(\frac{1}{75}\)

Step-by-step solution

Identify the integer pairs with a product of 36

To find the pairs of integers whose product equals 36, we must look for the factors of 36 that are within the set of integers 1 to 25. These pairs are: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6), (9, 4), (12, 3), (18, 2), and (36, 1). However, we need to discard the pairs that have elements outside the range of 1 to 25. This leaves us with the pairs (2, 18), (3, 12), (4, 9), and (6, 6).

Calculate the total number of pairs when selecting two numbers

To find the total number of pairs when selecting two numbers from a set of 25 integers, we can use the combination formula: C(n, k) = n! / (k!(n-k)!), where n represents the size of the set and k represents the number of elements you want to choose. In this case, n = 25 and k = 2: C(25, 2) = 25! / (2!(25-2)!) = (25*24) / 2 = 300.

Calculate the probability of a pair having a product of 36

We determined that there are 4 desired pairs with a product of 36 (out of the possible 300 pairs). So, the probability of selecting a pair with a product of 36 is given by: P(product = 36) = (number of desired pairs) / (total number of pairs) = 4 / 300 = 1 / 75. The answer is choice (4) \(\frac{1}{75}\).

Key concepts

Combinatorics in Mathematics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects following certain rules. While it might seem abstract, it's the backbone behind understanding complex problems in probability, such as the chance of drawing specific card combinations from a deck. An essential tool from combinatorics used in probability is the combinations formula, represented as C(n, k).

This formula is crucial in determining the number of ways we can select k items from a set of n discrete items, regardless of the order. The formula is articulated mathematically as: \[C(n, k) = \frac{n!}{k!(n-k)!}\]
In the context of the problem, we use the combinations formula to calculate the total possible pairs we can form when selecting two numbers from a set of up to 25 integers. Here, combinatorics simplifies the process to find the denominator in our probability calculation.

Factors and Multiples

Factors and multiples are fundamental concepts in arithmetic and number theory. A factor of a number is an integer that divides into that number without any remainder. Meanwhile, a multiple is a number that can be formed by multiplying the original number by an integer. Understanding these concepts becomes crucial when we're faced with problems that involve products of numbers, such as finding the probability of the numbers' product being a specific value.

For example, in our problem, to find the pairs whose product is 36, we list out all the factor pairs of 36. These pairs are multiple combinations of 36's factors, which we can then filter based on the given constraints—only considering integers between 1 to 25. Recognizing the valid factor pairs is a key step in determining the numerator for our probability calculation.

Probability Calculation

Probability calculation involves determining the chance of a particular event occurring compared to all possible outcomes. When we are talking about events where each outcome is equally likely, such as drawing balls from a bag or selecting random numbers, the probability can be calculated using the formula:\[P(A) = \frac{\text{Number of ways event A can occur}}{\text{Total number of possible outcomes}}\]
Applying this to our initial problem, we first identified all valid combinations where the product of two numbers is 36. Then, we calculated the total number of possible pairs of numbers that can be chosen from the range given. With this information, we determine the probability as the ratio of the number of successful outcomes (desired factor pairs) to the total number of possible outcomes (all number pairs). It's a direct application of the probability calculation to give us the final answer.