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Question

The letters of the word ZENITH are permuted and are arranged in an alphabetical order as in an English dictionary.

Then, the rank of the word ZENITH is ___


Solution

Let us arrange the letters of the word ZENITH in the alphabetical order.

EHINTZ

In the dictionary, the words appearing first will begin with E, then H, then I, then N , then T and at last Z.

Let us fix the first position as E.

E _ _ _ _ _  5P5 = 5! words

H _ _ _ _ _5P5 = 5! words

I _ _ _ _ _5P5 = 5! words

N _ _ _ _ _5P5 = 5! words

T _ _ _ _ _5P5 = 5! words

Z E H _ _ _3P3 = 3! words

Z E I _ _ _ 3P3 = 3! words

Z E N H _ _2P2 = 2! words

Z E N I H     1P1 = 1! words

Z E N I T H comes in order after all the above words

So, number of words before Z E N I T H = 5×5!+2×3!+2!+1 = 615

Since there are 615 words before ZENITH, rank of the word ZENITH = 616


Mathematics

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