The letters of the word ‘KUBER’ are written in all possible orders and arranged in a dictionary order. Find the rank of the work ‘KUBER’.

Hint:The given question requires us to find the rank of the work ‘KUBER’ when the word ‘KUBER’ is written in all possible orders and permutations and arranged in dictionary order alphabetically. So, we need to have thorough knowledge of concepts of permutations and combinations and also should know how to calculate factorials.

Complete step by step answer:
Number of alphabets in word ‘KUBER’$ = 5$.Now, we know that the total number of arrangements for the word consisting of n letters or alphabets is $n!$. So, the total number of arrangements of letters of the word ‘KUBER’ $ = 5! = 120$. Now, we need to arrange all the arrangements of letters of word ‘KUBER’ alphabetically so as to find the rank of the word ‘KUBER’ itself.

So, arranging the letters in alphabetical order, we get, B, E, K, R, U.
So, the number of words or arrangements starting with letter B $ = 4! = 24$
So, the number of words or arrangements starting with letter E $ = 4! = 24$
Now, the number of words or arrangements starting with KB $ = 3! = 6$
Now, the number of words or arrangements starting with KE $ = 3! = 6$
Now, the number of words or arrangements starting with KR $ = 3! = 6$

So, the last word arrangement starting with KR would be KRUEB.The next word arrangement after this would be KUBER. Hence, counting the number of words before ‘KUBER’, we get,total number of words before ‘KUBER’ appears in dictionary order is $66$.

So, the rank of the word ‘KUBER’ is 67 when the letters of the word ‘KUBER’ are written in all possible orders and arranged in a dictionary order.

Note:These kinds of questions which require us to find the rank of a work when the letters of the same word are arranged in dictionary order or alphabetical order require us to apply the concepts of permutations and combinations.