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September 27

Uses of Pascal's tetrahedron

The following is a use of Pascal's triangle:

To find how many ways there are to make a total of $n$ circles all black or red, the formula is just $2^{n}$ . For example, there are $2^{6}=64$ ways to make a group of 6 circles, all black or red, classified by whether each circle is black or red. An example is black-red-black-red-black-red.

But how about finding the number of ways to make a group of 6 circles, all black or red, classified by how many are black or red. To find out how many ways there are to make a total of $n$ circles all black or red classified by how many are black and how many are red, you use the n+1th row of Pascal's triangle. For $n=6$ , this means we use the seventh row, which is $1,6,15,20,15,6,1$ . This means that there is one way to color 6 circles where all of them are black, 6 where 5 are black and one is red, 15 where 4 are black and 2 are red, 20 where 3 are black and 3 are red, 15 where 2 are black and 4 are red, 6 where one is black and 5 are red, and one where all 6 are red.

How about a similar application for Pascal's tetrahedron?? Here is the seventh layer of the tetrahedron:

1 6 6 15 30 15 20 60 60 20 15 60 90 60 15 6 30 60 60 30 6 1 6 15 20 15 6 1

Just as the seventh row of Pascal's triangle can be used for the classification of ways to make 6 circles all of which are black or red classified by how many are black and how many are red, it is likewise true that the seventh layer of Pascal's tetrahedron can be used for... Georgia guy (talk) 14:06, 27 September 2024 (UTC)[reply]

It counts how many ways there are to make 6 circles all of which are black, red, or green, classified by how many are black, how many are red, and how many are green. For example, if you want there to be 2 of each colour, you get

{6 \choose 2,2,2}={6 \choose 2}{4 \choose 2}{2 \choose 2}=90

ways, which is exactly the middle entry of this layer. Double sharp (talk) 14:55, 27 September 2024 (UTC)[reply]

I would call 1, 6, 15, ... the sixth row, and the top row, with a single 1, the zeroth row. That's what Wikipedia's does in the articles I've seen at least. Anyway, this should be in the article Pascal's pyramid, and can be further extended to Pascal's simplex. The article on the Multinomial theorem is relevant here as well. --RDBury (talk) 17:25, 27 September 2024 (UTC)[reply]

RDBury, is the top row "1" not really a row?? Georgia guy (talk) 21:02, 27 September 2024 (UTC)[reply]

Well, you can call it whatever you want and index it as you like; I'm not going to get into the philosophy of rows. But the formulas are simpler if you start with the top "1" as row 0. --RDBury (talk) 00:55, 28 September 2024 (UTC)[reply]

RDBury, does that statement parallel the statement that trigonometry is simpler if you use radians as opposed to degrees?? Georgia guy (talk) 01:04, 28 September 2024 (UTC)[reply]

The OP was calling what most people call row 6 "the seventh row" and I thought that was worth pointing out for future reference. Getting caught up in what counts as a "row" and whether statements are parallel is a matter for philosophy and linguistics. --RDBury (talk) 02:20, 28 September 2024 (UTC)[reply]

@Georgia guy: Hopefully this looks more like what you intended...

             1
           6   6
         15  30  15
       20  60  60  20
     15  60  90  60  15
   6   30  60  60  30  6
 1   6   15  20  15   6   1

--CiaPan (talk) 13:32, 10 October 2024 (UTC)[reply]