Question

If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac.

Answer 1

In a dictionary the words at each stage are arranged in alphabetical order. In the given problem we must therefore consider the words beginning with a, b, c, d, e in order. 'a' will occur in the first place as often as there are ways of arranging the remaining 4 letteres all at a time i.e. 'a' will occur 4! times. Similarly b and c will occur in the first place the same number of times.

$\therefore$ Number of words starting with 'a' = 4!

= $4\times 3\times 2\times 1=24$

Number of words starting with 'b' = 4!

= $4\times 3\times 2\times 1=24$

Number of words starting with 'c' = 4!

= $4\times 3\times 2\times 1=24$

Number of words beginning with 'd' is 4!, but one of these words is the word debac.

So, we first find the number of words beginning with da,db,dc and dea

Number of words starting with da = 3! = 6

Number of words starting with db = 3! = 6

Number of words starting with dc = 3! = 6

Number of words starting with dea = 2! = 2

There are 2! words beginning with deb one of these words is the word debac itself

The first word beginning with deb is the word debac.

$\therefore$ Rank of debac = $3\times 24+3\times 6+2+1$

= 72+18+3

=90+3

=93