6. six men and four women are standing around a table that has four chairs. if four people are selected randomly to sit at the table, what is the probability that two men and two women will be selected
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To find the probability, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Step 1: Total number of possible outcomes
Since there are ten people in total, and four people are being selected, we can calculate the total number of ways to choose four people out of ten using combinations. This can be done using the formula:
nCr = n! / (r!(n-r)!)
Where n is the total number of objects (in this case, ten people) and r is the number of objects being selected (in this case, four people).
So, the total number of possible outcomes is:
10C4 = 10! / (4!(10-4)!)
= 10! / (4!6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
Therefore, there are 210 possible ways to select four people.
Step 2: Number of favorable outcomes
We want to select two men and two women. We need to calculate the number of ways to choose two men out of six, and two women out of four.
The number of ways to choose two men out of six is:
6C2 = 6! / (2!(6-2)!)
= 6! / (2!4!)
= (6 * 5) / (2 * 1)
= 15
Similarly, the number of ways to choose two women out of four is:
4C2 = 4! / (2!(4-2)!)
= 4! / (2!2!)
= (4 * 3) / (2 * 1)
= 6
Therefore, there are 15 ways to choose two men and 6 ways to choose two women.
Step 3: Calculating the probability
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
So, the probability of selecting two men and two women is:
(15 * 6) / 210
= 90 / 210
= 0.43 (rounded to two decimal places)
Therefore, the probability of selecting two men and two women is approximately 0.43 or 43%.