Harvey the Spider is putting on his argyle socks and boots in the morning. Naturally, he has 8 socks and 8 boots to put on, and for every one of his 8 legs, he has to put the sock on before the boot.

Let’s number Harvey’s feet 1 through 8, call the argyle sock for his foot , and call the boot for his foot . Then one valid order Harvey could use for putting on socks and boots is

Another valid order is

since each comes before its corresponding . But it would be impossible for Harvey to use the order

since now comes before . Harvey also can’t use the order

since all the sock-boot pairs are backwards.

Harvey is sleepy this morning and not paying attention to his order of socks and boots, so he puts on his socks and boots in random order. What is the probability that he picks a valid order for his socks and boots?

(If you get stuck, you might want to read the techniques in Better Ways to Count.)

The solution can be found here. Happy puzzling!

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