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

Solution:

In a dictionary, the words are listed and ranked in alphabetical order. In the given problem, we need to find the rank of the word ‘debac’.

For finding the number of words starting with a, we have to find the number of arrangements of the remaining 4 letters.

Number of such arrangements = 4!

For finding the number of words starting with b, we have to find the number of arrangements of the remaining 4 letters.

Number of such arrangements = 4!

For finding the number of words starting with c, we have to find the number of arrangements of the remaining 4 letters.

Number of such arrangements = 4!

For finding the number of words starting with d, fixing the next letter as a, we have to find the number of arrangements of remaining 3 letters.

Number of such arrangements = 3!

For finding the number of words starting with d, fixing the next letter as b, we have to find the number of arrangements of remaining 3 letters.

Number of such arrangements = 3!

For finding the number of words starting with d, fixing the next letter as c, we have to find the number of arrangements of remaining 3 letters.

Number of such arrangements = 3!

For finding the number of words starting with d, fixing the next letter as e:

First word- deabc

Second word- deacb

Third word- debac

Number of words after which we reach the word debac = 4!+4!+4!+3!+3!+3!+1+1+1 = 93

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