Consider all possible permutations of eight distinct elements $a, b, c, d, e, f, g, h$. In how many of them, will $d$ appear before $b$? Note that $d$ and $b$ may not necessarily be consecutive.
Asked
Active
Viewed 1,010 times
1
-
1Choose two positions. There are only one way to place d and b in those two positions so that d appears before b. – Lionel Ricci Feb 23 '16 at 15:41
-
Would there be a reason for $b$ to appear before $d$ more often than the other way around? Why (not)? – StackTD Feb 23 '16 at 15:42
-
1a personnal effort may be nice from ur end – Abr001am Feb 23 '16 at 15:54
-
Taking any permutation and putting d where b has been and b where d has been is a bijection with no fix-points that maps the subset "b before d" to "b after d". So the number of elements in those sets are equal and half the number of all permutations. – Gyro Gearloose Feb 23 '16 at 15:55
-
I did try have a solution I mean something that I think may be a solution. – Alex_ban Feb 23 '16 at 15:59
-
So what is that thing that you "think may be a solution"? – Rory Daulton Feb 23 '16 at 20:47
2 Answers
2
From these 8 elements we can create 8! permutations.
There is a one to one correspondence between the permutations where $d$ appears after $b$ and the permutations where $d$ appears before $b$. This is a simple switching of the elements $d$ and $b$.
Hence half of the 8! permutations will be so that $d$ appears before $b$. This equals 20160.
TCiur
- 498
-
those kinds of shortcuts always miss their road to my intuition, nice trick ! – Abr001am Feb 24 '16 at 09:45
-
I'm baffled. Can you assist pls? 1. How do you know anything about the "correspondence between the permutations where d appears after b and the permutations where d appears before b"? How do you know this is "one to one"? 2. Then how do you deduce "half of the 8! permutations"? – Jan 03 '22 at 06:58
-
0
- By inventing a function f(n) where n is the range of b
$f(n)=(n-1)!*\binom{8-2}{8-n}(8-n)!$ the possible permutations where b is at the $n$th rank.
This gives outcome to f(2)+f(3)+..f(8) all valid permutations .
Abr001am
- 746