Three distinct positive integers are selected at random from 9 consecutive positive integers. What is...

glitinosim3

glitinosim3

Answered

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.

Answer & Explanation

Charlee Gentry

Charlee Gentry

Expert

2022-07-04Added 19 answers

There will be three each of integers that are 0 , 1 , 2 mod 3 in the given range. The admissible selections sum to 0 mod 3, of which there are four ways to achieve it:
000   ( 1  selection )
111   ( 1 )
222   ( 1 )
012   ( 27 )
This gives 30 admissible selections out of ( 9 3 ) = 84 total selections, for a probability of 30 84 = 5 14

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