Many thanks to Sebastien Towers for contributing this week's puzzle.

Grace likes to have ketchup, mustard or mayonnaise in her sandwiches. She only ever uses exactly two of them at once.

The ketchup bottle, mustard jar and mayonnaise pot are lined up in that order in her fridge. When she makes a sandwich, she chooses two of them at random, takes them out to make a sandwich, and then puts them back. She always puts them back flipped round, so that e.g. if the order is Ketchup-Mustard-Mayo, and she takes out Ketchup and Mayo, she will put them back in so that the order is Mayo-Ketchup-Mustard.

If the order starts as Ketchup-Mustard-Mayo, and she makes 4 sandwiches, what is the probability that afterwards the order is still Ketchup-Mustard-Mayo?

What if she makes 7 sandwiches?

What if she makes n sandwiches?

Now suppose that at random one-third of the time she puts the condiments back in the right way round. How does that change your answer?