Since we know,
Permutation of n objects taking r at a time is nPr ,and permutation of n objects taking all at a time is n!
And,
We also know,
Permutation of n objects taking all at a time having p objects of the same type, q objects of another type, r objects of another type is \(\frac{n!}{p!\times q!\times r!}\).
i.e.,
The number of repeated objects of same type are in denominator multiplication with factorial.
Given,
The word ARRANGE. It has 7 letters, and it has 2 repeated letters ‘A’, ‘R.’ Of which, the letter A is repeated twice, and the letter R is also repeated twice. All other letters are distinct.
The problem can now be rephrased as to find a total number of permutations of 7 objects (A, R, R, A, N, G, E) of which two objects are of same type (A, A), and two objects are of another type (R, R).
Total number of such permutations = \(\frac{7!}{2!\times 2!}\)
= 1260
Hence,
A total number of permutations of the word ARRANGE is 1260.
