16 5 points According to the rules of a game if one lands in

16. (5 points) According to the rules of a game, if one lands in the dungeon then to escape one must roll a die until (j) the sum of the numbers rolled is at least 5 or (ii) some number has been rolled twice. If one keeps a record of the numbers that are rolled as they are rolled, how many different ways are there to escape from the dungeon.

Solution

Those with sum at least 5 are

1 throw: 2 ways [5 or 6]

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2 throws: 6^2 - n(4 or less) - n(those who have 5 or 6 as first term).

Those with 4 or less as sum, without repeating is (1,2), (1,3), (2,1), (3,1). [4 items]

Those with 5 or 6 as first roll have 2*6 =12 ways.

So,

2 throws: 36 - 4 - 12 = 20 ways

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For three throws, they must follow the 4 terms in 2 throws.

(1,2), (1,3), (2,1), (3,1)

Any next roll would actually finish it, so there are 4*6 = 24 ways to finish it in 3 rolls.

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Thus, there is a total of 2 + 20 + 24 = 46 [ANSWER, OPTION E]