Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:
Let A, B, and C be three boxes with three, four, and five cells, respectively. There are three yellow balls numbered 1 to 3, four green balls numbered 1 to 4, and five red balls numbered 1 to 5. The yellow balls are placed at random in box A, the green in B, and the red in C, with no cell receiving more than one ball. Find the probability that only one of the boxes will show no matches.
My question is more on how to interpret the problem. I actually cannot understand what is being asked and how was the experiment performed. Can anyone help me please? Also, if you have an answer, please explain the solution as well. Thanks
Best Answer
The only interpretation I can think of is that the cells are also numbered: in Box A they are $1,2,3$, in Box B $1,2,3,4$, in Box C $1,2,3,4,5$. Then a match is a ball numbered $i$ landing in a cell numbered $i$.
For a related single box problem, look up derangements.
The probability that Box A has no match is easy to find. Box B is a little harder, and for Box C the theory of derangements will greatly simplify the counting.