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Find the total number of permutations of the letters of each of the words given below: 

(i) APPLE 

(ii) ARRANGE 

(iii) COMMERCE 

(iv) INSTITUTE 

(v) ENGINEERING 

(vi) INTERMEDIATE

1 Answer

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To find: number of permutations of the letters of each word 

Number of permutations of n distinct letters is n! 

Number of permutations of n letters where r letters are of one kind, s letters of another kind, t letters of a third kind and so on = \(\frac{n!}{r!s!t!}\)

(i) Here n = 5

P is repeated twice 

So the number of permutations = \(\frac{5!}{2!}\) = 5 x 4 x 3 = 60

(ii) Here n = 7

A is repeated twice, and R is repeated twice

So, the number of permutations = \(\frac{7!}{2!2!}\) = \(\frac{7\times6\times5\times4\times3}{2}\) = 1260

(iii) Here n = 8 

M and E are repeated twice 

So, the number of permutations =   \(\frac{8!}{2!2!}\) = \(\frac{8\times7\times6\times5\times4\times3\times2}{4}\) = 10080

(iv) Here n = 9 

I is repeated twice, T is repeated thrice 

So, the number of permutations =  \(\frac{9!}{2!2!}\)  = 30240

(v) Here n = 11

E, N is repeated thrice, I,G are repeated twice 

So the number of permutations =   \(\frac{11!}{3!3!2!2!}\)  = 277200

(vi) Here n = 12

I and T are repeated twice, E is repeated thrice 

So, the number of permutations =   \(\frac{12!}{2!2!3!}\)  = 19958400

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