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Permutation Of Objects When Few Are Identical - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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• 82 Questions around this concept.

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Consider the word "ENGINEERING". In how many ways can the letters be arranged such that all the vowels always come together, but the two 'E's are not adjacent and the three 'N's are in alphabetical order?

How many different words can be formed with the letters of the word "ENCAPSULATE" such that: (i) Four vowels occupy the odd places? (ii) The word begins with the letter N? (iii) The word begins with the letter E and ends with the letter T?

In how many ways can the letters of the word "APPRECIATE" be arranged such that the word should have the first letter P?

How many ways can the letters of the word "ARRANGE" be arranged such that the two R's are never together?

In how many ways can the letters of the word "PERMUTE" be arranged if all the letters must be used?

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Concepts Covered - 1

PERMUTATION OF OBJECTS WHEN FEW ARE IDENTICAL

If there n objects of which p objects are of one type, q objects of another type, r objects of yet another type and all others are distinct, the total number of ways of arranging all the objects is $\\\mathrm{\frac{n!}{p!q!r!}}$

Proof:

Suppose the total number of permutations are x, now if we replace all p identical objects by p different objects then we have x × p! arrangements. The number of arrangements, if we do the same thing with q and r, will be x × p!×q!×r!

Now, we have replaced all identical objects and we are left with n different object which can be arranged in n! Ways.

Hence, x × p!×q!×r! = n!

So, $\\\mathrm{x=\frac{n!}{p!q!r!}}$

Example: In how many ways can the letters of the word “MISSISSIPPI” be arranged?

Solution: repeated letters I = 4 times, S = 4 times and P=2 times

So using the above formula we have $\\\mathrm{x=\frac{11!}{4!\times4!\times2!}}$

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PERMUTATION OF OBJECTS WHEN FEW ARE IDENTICAL

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Books

Reference Books

PERMUTATION OF OBJECTS WHEN FEW ARE IDENTICAL

Mathematics Textbook for Class XI

Page No. : 142

Line : 1