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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
GARDEN
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Solution
The letters of the word arranged in the dictionary order is A, D, E, G, N, R
Total number of letters = 6
The number of words begins with A = 5!
The number of words begins with D = 5!
The number of words begins with E = 5!
The number of words beginning with G = 5!
(But one of these words is GARDEN)
The number of words beginning with GAD = 3!
The number of words beginning with GAE = 3!
The number of words beginning with GAN = 3!
There are 3 ! words beginning with GAR one of these words is GARDEN.
The first word beginning with GAR is the word GARDEN.
∴ The rank of the word GARDEN 3 × 120 + 3 × 6 + 1 = 360 + 18 + 1 = 379
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