Discrete Math HELP!?

Four cards, numbered 1, 2, 3, and 4, are to be placed in four boxes numbered 1, 2, 3, 4; one card in each box. Define a sample space of ordered quadruples that indicate the card in each slot. (For example, 1243 indicates card 1 in box 1, card 2 in box 2, card 4 in box 3, and card 3 in box 4.) Using the sample space, calculate the probability that 

(a) exactly two card numbers and two box numbers coincide,

(b) at least two card numbers and two box numbers coincide, and

(c) no card number coincides with its box number

1 Answer

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  • 1 month ago

    Sample space (has 24 elements) :

    1234 ……… 2134 …….. 3124 …….. 4123

    1243 ……… 2143 …….. 3142 …….. 4132

    1324 ……… 2314 …….. 3214 …….. 4213

    1342 ……… 2341 …….. 3241 …….. 4231

    1423 ……… 2413 …….. 3412 …….. 4312

    1432 ……… 2431 …….. 3421 …….. 4321

    (a) exactly two card numbers and two box numbers coincide:

    1243, 1324, 1432, 2134, 3214, 4231 ……….. (6 elements)

    probability = 6/24 = 1/4 = 0.25

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    (b) at least two card numbers and two box numbers coincide

    1234, 1243, 1324, 1432, 2134, 3214, 4231…….. (7 elements)

    probability = 7/24 ≈ 0.292

    -----

    (c) no card number coincides with its box number

    2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321 …. (9 elements)

    probability = 9/24 = 3/8 = 0.375

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