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counting and probability intermediate

Problem

Mary and James each sit in a row of 7 chairs. They choose their seats at random. What is the probability that they don't sit next to each other?
Solution
There are ways in which Mary and James can choose 2 chairs, if we don't worry about the order in which they sit. Although we can use casework to count the number of ways they can choose chairs which are not next to each other, it is easier to use complementary counting. If we number the chairs in order, then there are 6 ways Mary and James can choose chairs next to each other: they can sit in the first two chairs, or chairs and or chairs and etc., up to chairs and Therefore and therefore
Final answer
\frac{5}{7}