# A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G(for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?

Given the molecules’ initials A, G, T, and C (All are repeated thrice).

AAAGGGTTTCCC

To find: Number of arrangements of these 12 molecules in such a way that all arrangements must be distinct.

The problem can now be rephrased as to find a number of permutations of 12 objects in which 3 objects are of one type, 3 objects are of another type, 3 objects are of a third type, and remaining 3 objects are of different type.

Since we know, Permutation of n objects taking r at a time is nPr,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.

The number of permutation of 12 objects with repeating molecules in the factor of 3

= 369600

Hence, total number of permutation of given 12 molecules will be equals to 369600.

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