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10 Questions 10 Marks 10 Mins
Concept:
Let there be n things of which p1 are alike of one kind, p2 are alike of another kind, p3 are alike of 3rd kind, ..…, pr are alike of rth kind such that p1 + p2 + ….+ pr = n. Then the permutations of n objects is \(\rm \frac{n!}{(p_1!)\times (p_2!)\times ....\times (p_r!)} \)
Calculation:
The word ALLAHABAD contains 9 letters, in which A occur 4 times, L occurs twice and the rest of the letters occur only once.
Number of different words formed by the word ALLAHABAD using all the letters
\(\rm =\frac{9!}{4!\times2!}=\frac{9\times8\times7\times6\times5\times4!}{4!\times2}\)
\(=\rm \frac{72\times7\times30}{2}\)
= 7560
Now, let us take both L together and consider (LL) as 1 letter
Then we will have to arrange 8 letters, in which A occurs 4 times and the rest of the letters occur only once
So, the number of words having both L together \(=\rm \frac{8!}{4!}=\frac{8\times7\times6\times5\times4!}{4!}\)
= 1680
Hence, the number of words with both L not occurring together
= 7560 - 1680
= 5880
Hence, option (3) is correct.
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How many different words can be formed by using the letters of the word ‘ALLAHABAD?
(a) In how many of these do the vowels occupy even positions.
(b) In how many of these, the two L’s do not come together?
Number of letters in word 'ALLAHABAD' = (A → 4, L → 2, H → 1, B → 1, D → 1) = 9
Number of arrangements =
They can occupy even places (2, 4, 6, 8) in
Number of permutations =
= 7560 - 1680 = 5880.
HI, thanks for this wonderful stuff. I have a concern with last part of this question where you multiplied 1680 by saying both L can be arranged in two ways by themselves. Is it correct if both is the same word? Kindly reply if you really have started this forum for the visitors, not for the sake of "Link Building for search engines"
Best Regards