Worked solution
Separate repeated-ball selections into disjoint cases
A box contains 8 balls, of which 3 are identical (and so are indistinguishable from one another) and the other 5 are different from each other. 3 balls are to be picked out of the box, the order in which they are picked out does not matter. Find the number of different possible selections of 3 balls.
[2006/I/4]
CJC · Discussion · Question 9
Answer
26 selections.
Three identical balls
Name the identical type I and the other balls A, B, C, D and E. Selecting I, I, I gives 1 possible selection.
Exactly two of type I
There is 1 way to take two indistinguishable I balls and choices for the remaining distinct ball. This gives 5 selections.
Three different types
Choose three types from A, B, C, D, E and I: . This combines the cases of zero or one I without overlap. The full total is .
Common mistakes
The three-different-types case may include one I. Counting it again separately would duplicate selections.
Solution notes
All printed asks, working, counting factors and source annotations visually compared and retained. No marks are printed for this question.