Probability question regarding random assignment

Question

Let's say there are 24 study abroad students in Europe, and they are to be placed in host homes in the U.S. in groups of 3. The assignments will be determined by a lottery system, where the names are randomly drawn to determine which of the 8 host homes the students will stay at. Let's say there are 3 siblings in that group of 24, and they are really hoping they get placed in the same home. What are the chances that this will occur?

Answer 1

The first thing to do here is calculate how many combinations of 3 students we can get from a group of 24 students by using the combination formula:

C(n, r) = n! / ((n - r)! × r!)

There are C(24, 3) possible combinations of 3 students out of the total of 24

C(24, 3) = 24! / ((24 - 3)! × 3!) = 2024

Out of our 2024 possible combinations of 3 students out of 24, only 1 combination has all 3 siblings in it. The probability that all 3 siblings are selected in the same group of 3 is 1 / 2024 = 0.000494. The question says there are 8 homes, but it doesn't state that it matters which of they 8 homes they get placed in, so the probability remains 0.000494 or 0.0494%.