"The organizer of a television show must select 5 people to participate in the show. The participants will be selected from a list of 21 people who have written in to the show. If the participants are selected randomly, what is the probability that the 5 youngest people will be selected?"
Mathematics · College · Mon Jan 18 2021
Answered on
To determine the probability that the 5 youngest people will be selected, we need to understand the concept of combinations. A combination is a selection of items from a larger pool where the order of selection does not matter. The number of possible combinations of selecting r items from a set of n items is given by the combination formula:
nCr = n! / [(n - r)! * r!]
where "n!" represents the factorial of n, which is the product of all positive integers up to n (for instance, 5! = 5×4×3×2×1).
Let's calculate the total number of ways to choose 5 people from a list of 21 people, which will act as the denominator in our probability calculation.
Total ways to choose 5 people out of 21 (21C5): 21C5 = 21! / [(21 - 5)! * 5!] = 21! / (16! * 5!)
Now we calculate the number of ways to choose the 5 youngest people. Since we are specifically choosing a predefined set of 5 people, there is only one way to do this. So, this will be the numerator of our probability calculation.
Ways to choose the 5 youngest people = 1.
Now, let's calculate the probability: Probability = (Number of ways to choose the 5 youngest people) / (Total number of ways to choose 5 people) = 1 / 21C5 = 1 / (21! / (16! * 5!))
Without doing the calculation, we can simplify the expression because a lot of factorials will cancel out.
Probability = 1 / [ (21 × 20 × 19 × 18 × 17) / (5 × 4 × 3 × 2 × 1) ] = (5 × 4 × 3 × 2 × 1) / (21 × 20 × 19 × 18 × 17)
Do the math:
Probability = 1 / (21 × 20 × 19 × 18 × 17 / 120) = 120 / (21 × 20 × 19 × 18 × 17)
We can simplify further:
Probability = 1 / (21 × 4 × 19 × 18 × 17) = 1 / (287304) approximately.
This probability is extremely small, which makes sense as there is a very low chance of randomly selecting the 5 youngest people out of 21.