glitinosim3

2022-07-03

Three distinct positive integers are selected at random from 9 consecutive positive integers. What is the probability that their average is also an integer?
Is there any way to find it efficiently? I'd taken example of 1,2,3,4,5,6,7,8,9 then there are a lot of combinations to give a number divisible by three which give an integer average value. It's difficult to find the way :D
Thank you.

Charlee Gentry

Expert

There will be three each of integers that are $0,1,2mod3$ in the given range. The admissible selections sum to $0mod3$, of which there are four ways to achieve it:

This gives 30 admissible selections out of $\left(\genfrac{}{}{0}{}{9}{3}\right)=84$ total selections, for a probability of $\frac{30}{84}=\frac{5}{14}$

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