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You are going to build a tower with coloured blocks. There are ten available blocks, of which three are white, two are red, two are yellow, one is green, one is blue and one is black. The tower you are going to build will be ten blocks high.

How many different towers, with regards to colour, can be built?

My attempt to solve the problem is as follows:

There is a total of 10 blocks and 6 different colours. The colours can be divided into groups by $(6 - 1) = 5$ separators. There will be a total of 15 positions on which the 5 separators can be distributed. The number of different towers (in terms of colour) that can be built are the same as the number of ways that the separators can be distributed, which is $15 \choose 5$ $=3003$ different ways.

Is this the correct answer and a correct way of reasoning?

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  • $\begingroup$ The idea is not right. And currently the blocks available are incompletely described, you have identified only $8$. $\endgroup$ – André Nicolas May 20 '15 at 21:02
  • $\begingroup$ Sorry, forgot to mention the two yellow ones. Post has been edited :) Well, how to solve it, if the idea isn't right? :/ $\endgroup$ – Kaedos May 20 '15 at 21:04
  • $\begingroup$ The way to solve it is described in the answer by CuddlyCuttlefish. $\endgroup$ – André Nicolas May 20 '15 at 21:07
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Your instructor is likely assessing your ability to recognize that this is equivalent to problems like counting the number of distinct words that can be written using all of the letters in 'MISSISSIPPI'.

Apply identical reasoning to this situation: $$\frac{10!}{3!2!2!1!1!1!}$$

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  • $\begingroup$ Let me get this straight... $10!$ means the ways that all of the blocks can be organized. By dividing with for instance $3!$, one removes all the towers where all the red blocks are arranged in the same way. Correct? $\endgroup$ – Kaedos May 20 '15 at 21:07
  • $\begingroup$ ...and by dividing with $3!2!2!1!1!1!$, one gets only uniquely coloured towers? $\endgroup$ – Kaedos May 20 '15 at 21:08
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    $\begingroup$ Yes, that is correct; you're trying to remove towers that are 'distinct' due to a permutation of indistinguishable blocks. $\endgroup$ – TokenToucan May 20 '15 at 21:09

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