#Probability word problem

19 messages · Page 1 of 1 (latest)

abstract sail
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I stumbled across this word problem and am confused about the answer. The answer uses a table providing the different cases of exact number of matches and the corresponding possible ways to achieve the number of matches.

I tried solving for the number of ways for one match and I know that the 5 choose 1 combination comes from having only 1 match but I don't fully understand why there are 9 ways to arrange the 4 other cards.

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untold tuskBOT
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untold tuskBOT
slow sphinx
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But also, you know already there's only 9 options, you can just write out all card-orders that match in the 1st slot

abstract sail
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true

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I just thought there was a way without brute force

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But it makes sense why they made a table then

slow sphinx
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I mean, it should be clear why 4 gives 0 for example

abstract sail
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yea

slow sphinx
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But here's one way to think of it w/o brute force (which, I wont lie, came from me just writing out some options)

We can assume the match is in I
What happens if the 2nd card is X? (Note that the 2nd card cant be V, so by considering X, I will multiply the answer by 3 since they're all symmetric)
Well then my sequence is I X ? ? ? and the cards V C L must be placed in the ?'s so that there's no matches with X L C.
Clearly the 3rd card is free to be whatever, but that choice forces the other 2
(If the 3rd card is V, the remaining order must be C L else you get matches. If it's C, then it must V L, etc)

slow sphinx
# abstract sail yea

so there's 3 options for the 2nd card (X C L), 3 options for one of the remaining cards, and the last 2 are picked by not allowing matches.

abstract sail
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but the logic does check out

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tyy

abstract sail
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+close

untold tuskBOT
# abstract sail +close
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untold tuskBOT
# untold tusk

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slow sphinx