Chant6j

2022-07-02

How many ways can you pick exactly the same 4 people from a group of 20
The question is what is the probability that Barbara, Carl, Georg, and Henrietta are randomly chosen from a set of twenty clients for a four-person group date if all possible choices are equally likely?
I know the formula is P = Favorable outcomes / possible outcomes. For whatever reason I am not sure how to get the numerator. The possible outcomes is $\left(\genfrac{}{}{0}{}{20}{4}\right)$, but the numerator is alluding me. It is not a multiplication rule because it doesn't allow for repeating, it can't be a permutation because order doesn't matter, leaving a combination I think. but then I did $\left(\genfrac{}{}{0}{}{20}{4}\right)$.
I got $\frac{116280}{4845}$, which is a number larger than 1.
What method am I missing? How do I get the number of ways I can exactly pick the Barbara, Carl, Georg, and Henrietta?

Oliver Shepherd

Expert

Step 1
Let us do it by using combinations: There are 4845 of picking 4 people from 20 (order doesn't matter). Out of those 4845, there is only one possibility of picking Barbara, Carl, Georg and Henrietta.
Step 2
Therefore, there is a $\frac{1}{4845}$ chance that those 4 will be picked.

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