Problem 22 · 2022 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
Veronika wears five rings as shown. How many different ways are there for her to take off the rings one by one?
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Answer: B — 20
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Hint 1 of 2
Removing rings is just ordering 5 items, but a stacked ring can only leave after the ring above it.
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Hint 2 of 2
Count all 5! orders, then divide out the orders forbidden by the one stack of three.
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Approach: count orderings, then divide by the forced order within the stack
- Ignoring constraints there are \(5! = 120\) ways to pick an order to remove the five rings.
- The three rings stacked on one finger must come off top-to-bottom, which is just 1 of their \(3! = 6\) possible internal orders.
- So the count is \(120 / 6 = 20\) valid orders, answer B.
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