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There are 4 cards. One is labeled 12, a second 34, a third 56, and the last 78. They are lined up at random. The probability that exactly 4 digits will appear in a consecutive, increasing order is m/n in simplest form. What is m+n? (Note that a possible correct ordering is 12 34 78 56. An incorrect one is 12 78 56 34 since, while 12 78 is in increasing order, the numbers are not consecutive.)

 Dec 16, 2020
 #1
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There are only 4! = 24 permutations, which are listed here. Simply count the ones that meet your conditions and divide that number by the total of 24:

 

12,34,56,78
12,34,78,56
12,56,34,78
12,56,78,34
12,78,34,56
12,78,56,34
34,12,56,78
34,12,78,56
34,56,12,78
34,56,78,12
34,78,12,56
34,78,56,12
56,12,34,78
56,12,78,34
56,34,12,78
56,34,78,12
56,78,12,34
56,78,34,12
78,12,34,56
78,12,56,34
78,34,12,56
78,34,56,12
78,56,12,34
78,56,34,12
#N : 24

 Dec 16, 2020
 #2
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Thank you !

Guest Dec 16, 2020

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