Probability of No Woman Selected

C

Probability of both positions being filled by men = 6/10*5/9 = 1/3.

To find the probability that no woman is selected for the two executive positions, we need to calculate the probability of selecting two men from the available pool of candidates.

We are given that there are four women and six men competing for the two executive positions. We can use combinations to calculate the number of ways to select two men from the pool of six men.

The number of ways to select two men from six men can be calculated using the combination formula:

C(n, r) = n! / (r! * (n - r)!)

Where n is the total number of items and r is the number of items to be chosen.

In this case, we have six men and we want to choose two, so the calculation becomes:

C(6, 2) = 6! / (2! * (6 - 2)!)

C(6, 2) = (6 * 5 * 4!) / (2! * 4!)

C(6, 2) = (6 * 5) / (2 * 1)

C(6, 2) = 15

Therefore, there are 15 different ways to select two men from the pool of six men.

Now, let's consider the total number of ways to select any two candidates from the pool of ten (four women and six men):

C(10, 2) = 10! / (2! * (10 - 2)!)

C(10, 2) = (10 * 9 * 8!) / (2! * 8!)

C(10, 2) = (10 * 9) / (2 * 1)

C(10, 2) = 45

Therefore, there are 45 different ways to select any two candidates from the pool of ten.

To find the probability that no woman is selected, we divide the number of ways to select two men by the total number of ways to select any two candidates:

Probability = Number of ways to select two men / Total number of ways to select any two candidates

Probability = 15 / 45

Probability = 1/3

Hence, the probability that no woman is selected for the two executive positions is 1/3. Therefore, the correct answer is option C: 1/3.