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A child has 12 blocks, 6 black, 4 red, 1 blue, 1 white. How many ways can he arrange them in a chain?

I am not quite sure how to approach this problem. Sequential reasoning doesn't work(as far as I can see) since there are different numbers of blocks of different colors.

Any hints would be appreciated!

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    Does the reflection of a chain count as distinct? – 2017-02-22
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    By reflection, do you mean inverted order? Then yes. – 2017-02-22

2 Answers 2

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Suppose the blocks have distinct numbers on them. Then obviously there are $12!$ arrangements. Now successively erase numbers on the black, red, blue and white blocks in order; at each stage this causes groups of $6!$, $4!$, $1!$ and $1!$ distinct arrangements to become one, dividing the count of arrangements by the respective number. Thus the number of arrangements of the original blocks is $12!/(6!4!1!1!)=27720$.

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You can see your chain of blocks as a word composed of $12$ letters, in which one letter appears $6$ times and another $4$ times. Thus the number of ways you can arrange them in a chain equals the number of anagrams of the word which is $\frac {12!}{6!4!}=27720$

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    I don't see why rephrasing the question so it doesn't involve blocks makes the answer any clearer. – 2017-02-22
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    I actually find it easier to solve a problem in terms of its standard form – 2017-02-22