Q. 55.0( 2 Votes )

# How many words can be formed with the letters of the word ‘PARALLEL’ so that all L’s do not come together?

Answer :

Given, the word PARALLEL. It has 8 letters of which 2 letters (A, L) are repeating. The letter A is repeated twice, and the letter L is repeated thrice in the given word PARALLEL.

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

How are we going to find it? First, we find all arrangements of word PARALLEL and then we minus all those arrangements of word PARALLEL in such a way that all L’s do come together, from it. This will exactly be the same as- all number of arrangements such that not all L’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 PARALLEL: Total letters 8. Repeating letters A and L, the letter A repeating twice and letter L repeating thrice. The total number of arrangements

Now we find a total number of arrangements such that all L’s do time 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 L, L, L) are 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 L, L, L).

Letters in word PARALLEL: 8 letters

Letters in a new word: LLL, P, A, A, R, E: 6 letters (Letter A repeated twice).

Total number of word arranging all the letters

where second fraction 3! divided by 3! comes from arranging letters inside the group LLL: arrangements of three letters where all the three letters are same = 1 (Obviously! You can even think of it).

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

= 3000

Hence, a total number of arrangements of word PARALLEL in such a way that not all L’s do come together equals to 3000.

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Which of the following are true :

(2 + 3)! = 2! + 3!

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