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Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris\textquotesingle s. What is the expected number of balls that occupy their original positions after these two successive transpositions? 
Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability
After the first swap, we do casework on the next swap. Case 1: Silva swaps the two balls that were just swapped There is only one way for Silva to do this, and it leaves 5 balls occupying their original position. Case 2: Silva swaps one ball that has just been swapped with one that hasn\textquotesingle t swapped There are two ways for Silva to do this, and it leaves 2 balls occupying their original positions. Case 3 : Silva swaps two balls that have not been swapped There are two ways for Silva to do this, and it leaves 1 ball occupying their original positions. Our answer is the average of all 5 possible swaps, so we get $$ \frac{5+2 \cdot 2+2 \cdot 1}{5}=\frac{11}{5}=(\text { D) } 2.2 $$ 