HELP? Consider the 720 permutations of the numbers 1, 2, 3, 4, 5, 6. In how many of these permutations does 1 appear next to 2 or 3 appear next to 4 or 5 appear next to 6?

Step-by-step explanation:

The set {1,2,3,4,5,6} has a total of 6! permutations

a. Of those 6! permutations, 5!=120 begin with 1. So first 120 numbers would contain 1 as the unit digit.

b. The next 120, including the 124th, would begin with  '2'  

 c. Then of the 5! numbers beginning with 2, there are 4!=24 including the 124th number, which have the second digit =1

d. Of these 4! permutations beginning with 21, there are 3!=6 including the 124th permutation which have third digit 3

e. Among these 3! permutations beginning with 213, there are 2 numbers with the fourth digit =4 (121th & 122th), 2 with fourth digit 5 (numbers 123 & 124) and 2 with fourth digit 6 (numbers 125 and 126).

Lastly, of the 2! permutations beginning with 2135, there is one with 5th digit 4 (number 123) and one with 5 digit 6 (number 124).

∴   The 124th number is 213564

Similarly reversing the above procedure we can determine the position of 321546 to be 267th on the list.