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May 10, 2024# Question 10377 – Combinatorics

The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is _______.

Correct Answer: A

Question 1 Explanation:

There are 5 places.

― ― ― ― ―

Given: L I L A C

The derangement formula ⎣n!/e⎦ cannot be directly performed as there are repeated characters.

Let’s proceed in regular manner:

The L, L can be placed in other ‘3’ places as

(1) Can be arranged such that A, I, C be placed in three positions excluding ‘C’ being placed at its own position, which we get only 2×2×1 = 4 ways.

― ― ― ― ―

Given: L I L A C

The derangement formula ⎣n!/e⎦ cannot be directly performed as there are repeated characters.

Let’s proceed in regular manner:

The L, L can be placed in other ‘3’ places as

(1) Can be arranged such that A, I, C be placed in three positions excluding ‘C’ being placed at its own position, which we get only 2×2×1 = 4 ways.

Similarly (2) can be filled as A, I, C being placed such that 4th position is not filled by A, so we have 2×2×1 = 4 ways. Similarly with (3).

Totally, we get 4+4+4 = 12 ways.

12

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