# A linear chain is made of 20 identical links. Each link can be made in 7 different colors. How many  physically  different chains are there?

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Feb 7, 2018

For each of 20 links, there are 7 choices, each time the choice is independent of previous choices, so we can take product.

Total number of choices = $7 \cdot 7 \cdot 7. . . \cdot 7 = = {7}^{20}$

But since chain can be reversed, we need to count distinct sequences.

First, we count number of symmetric sequences: i.e last 10 links take the mirror image of first 10 links.

Number of symmetric sequences = number of ways so select first 10 links = ${7}^{10}$

Except for these symmetric sequences, the non symmetric sequences can be reversed to produce a new chain. This means that only half of non-symmetric sequences are unique.

Number of unique sequences = (Number of non-symmetric)/2 + Number of symmetric sequences

$= \frac{{7}^{20} - {7}^{10}}{2} + {7}^{10} = 39896133290043625$

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