Answer :

Given, the word ARRANGE. It has 7 letters of which 2 letters (A, R) are repeating. The letter A is repeated twice, and the letter R is also repeated twice in the given word.

To find: Number of ways the letters of word ARRANGE be arranged in such a way that not all R’s do come together.

First, we find all arrangements of word ARRANGE, and then we minus all those arrangements of word ARRANGE in such a way that all R’s do come together, from it. This will exactly be the same as- all number of arrangements such that not all R’s do come together.

Since we know, Permutation of n objects taking r at a time is ^{n}P_{r},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 . i.e. the, number of repeated objects of same type are in denominator multiplication with factorial.

A total number of arrangements of word ARRANGE: Total letters 7. Repeating letters A and R, the letter A repeating twice and letter R repeating twice. The total number of arrangements

Now we find a total number of arrangements such that all R’s do come together.

A specific method is usually used for solving such type of problems. According to that, we assume the group of letters that remain together (here R, R) is assumed to be a single letter and all other letters are as usual counted as a single letter. Now find a number of ways as usual; the number of ways of arranging r letters from a group of n letters is equals to ^{n}P_{r}. And the final answer is then multiplied by a number of ways we can arrange the letters in that group which has to be stuck together in it (Here R, R).

Letters in word ARRANGE: 7 letters

Letters in a new word: A, RR, A, N, G, E: 6 letters (Letter A repeated twice).

Total number of word arranging all the letters

where second fraction 2! divided by 2! comes from arranging letters inside the group RR: arrangements of two letters where all the two letters are same = 2!/2! = 1 (Obviously! You can even think of it).

Now, a Total number of arrangements where not all R’s do come together is equals to total arrangements of word ARRANGE minus the total number of arrangements in such a way that all R’s do come together.

= 900

Hence, the total number of arrangements of word ARRANGE in such a way that not all R’s come together is equals to 900.

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