Two distinct numbers are picked randomly from the first ten natural numbers. Find the probability that the L.C.M of the two numbers is 6.

To find the probability that the least common multiple (L.C.M) of two distinct numbers chosen from the first ten natural numbers is 6, we first need to identify the pairs of numbers that meet this condition.

Understanding L.C.M of 6

The L.C.M of two numbers is the smallest number that is a multiple of both. For the L.C.M to be 6, the pairs must consist of numbers whose multiples include 6. The factors of 6 are 1, 2, 3, and 6.

Valid Pairs

The pairs of distinct numbers from the first ten natural numbers (1 to 10) that have an L.C.M of 6 are:

• (2, 3)
• (3, 2)
• (1, 6)
• (6, 1)
• (2, 6)
• (6, 2)

Total Possible Pairs

Next, we calculate the total number of ways to choose 2 distinct numbers from 10. This can be done using the combination formula:

C(n, r) = n! / [r!(n - r)!]

For our case, it is:

C(10, 2) = 10! / [2!(10 - 2)!] = 45

Calculating Probability

Now, we have 6 valid pairs that yield an L.C.M of 6 and a total of 45 possible pairs. The probability is calculated as:

Probability = (Number of favorable outcomes) / (Total outcomes)

Thus, the probability is:

Probability = 6 / 45 = 2 / 15

Final Result

The probability that the L.C.M of two distinct numbers chosen from the first ten natural numbers is 6 is 2/15.