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 :DThank you.

Question

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&#039;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&#039;s difficult to find the way :DThank you.

Charlee Gentry · Accepted Answer

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