S = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} is a set of 10 natural numbers. Suppose, 6 numbers are chosen from the set at random. What is the probability that equal number of even and odd numbers are chosen?
To find the probability that an equal number of even and odd numbers are chosen, we need to calculate the number of favorable outcomes (where the chosen numbers have an equal number of even and odd numbers) and the total number of possible outcomes (all the combinations of choosing 6 numbers from the given set).
First, let's count the number of favorable outcomes. In order to have an equal number of even and odd numbers, we can choose 3 even numbers and 3 odd numbers, or 2 even numbers and 2 odd numbers, with a total of 6 numbers chosen.
Option 1: Choosing 3 even numbers and 3 odd numbers
There are 5 even numbers in the given set: {4, 16, 36, 64, 100}.
We need to choose 3 numbers from this set, so the number of ways to choose 3 even numbers is 5C3, which is calculated as (5!)/(3!(5-3)!) = 10.
Similarly, there are 5 odd numbers in the given set: {1, 9, 25, 49, 81}.
We need to choose 3 numbers from this set, so the number of ways to choose 3 odd numbers is 5C3, which is also 10.
Therefore, the number of favorable outcomes for option 1 is 10 * 10 = 100.
Option 2: Choosing 2 even numbers and 2 odd numbers
We can choose 2 even numbers from the available 5 even numbers, which is 5C2 = 10.
Similarly, we can choose 2 odd numbers from the available 5 odd numbers, which is also 5C2 = 10.
Therefore, the number of favorable outcomes for option 2 is 10 * 10 = 100.
Now, let's calculate the total number of possible outcomes, which is the number of ways to choose 6 numbers from the given set of 10 numbers.
The total number of possible outcomes is 10C6 = (10!)/(6!(10-6)!), which simplifies to 210.
To find the probability, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (100 + 100) / 210
= 200 / 210
= 20 / 21
So, the probability that an equal number of even and odd numbers are chosen is 20/21.