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Question
If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
DANGER
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Solution
The dictionary order of the letters of the given word is A, D, E, G, N, R
In the dictionary order of words which begin with A, comes first.
If we fill the first place with A, the remaining 5 letters can be arranged in 5! ways. Proceeding like this
Number of words beginning with D = 5! = 120
Number of words beginning with DAE = 3! = 6
Number of words beginning with DAG = 3! = 6
Number of words beginning with DANE = 2! = 2
Number of words beginning with DANGE = 1! = 1
(which is the word DANGER)
∴ The rank of the word DANGER = 120 + 6 + 6 + 2 + 1 = 135
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