# 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

Relevance
• 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

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(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