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Question: How many words can be formed with the letters of the word ‘PARALLEL’ so that all L’s do not come tog...

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

Explanation

Solution

In this problem it is difficult to find the cases. So that all L’s do not come together in the word ‘PARALLEL’. So, we will find the cases in which two ‘L’ come together and then subtract it from all the total possible arrangements of all 88 letters of word ‘PARALLEL’.

Complete Step-by-step Solution
We have given a word ‘PARALLEL’ here and we have to find all the words that can be formed in which all the ‘L’ do not come together.
Now, take the word ‘PARALLEL’. In this word, ‘L’ comes three times and ‘A’ comes two times.
We divided 8!8! by 2!×3!2! \times 3! because ‘A’ comes twice and ‘L’ comes twice in the word
Now, we will find all the possible arrangements of the word parallel.
So, the possible arrangements are:-
=8!2!×3!= \dfrac{{8!}}{{2! \times 3!}}
Now, solve this for the possible arrangements.
=8×7×6×5×4×3×2×12×1×3×2×1 =3360  = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times {{3 \times 2 \times 1}}}}{{2 \times 1 \times {{3 \times 2 \times 1}}}} \\\ = 3360 \\\
Similarly, we divided 6!6! by 2!2! in the second case.
We have found the total number of cases and then subtracted the number of cases in which two ‘L’ come together from it.
Now, we will find the cases in which two ‘L’ come together.
Let us assume two ‘L’ as a single letter.
Now, we have six letters having 22’L’.
Total Possible arrangementsof this case is =6!21 = \dfrac{{6!}}{{21}} .
Now, solve this.
=6×5×4×3×2×12×1 =360  = \dfrac{{6 \times 5 \times 4 \times 3 \times {{2 \times 1}}}}{{ {{2 \times 1}}}} \\\ = 360 \\\
So, the total possible arrangements, in this case, are 360360.
So, we will subtract 360360 from the33603360 to find the total cases in which all ‘L’ do not come together.
So, the required possible arrangements are:-
=3360−360 =3000  = 3360 - 360 \\\ = 3000 \\\
Hence, 30003000 words can be formed.

Note:
In this type of question analyze the shape of the object given to know its capacity, whether it is a sphere of cuboids or cubical or cone look for volume formula. Find volume always where capacity is asked according to the given dimensions.