Wikipedia:Reference desk/Archives/Mathematics/2024 August 4

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August 4

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Even binomial coefficients

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Is there an elementary way to show that all the entries except the outer two, on a row of Pascal's triangle corresponding to a power of 2, are even? 2A00:23C6:AA0D:F501:658A:3BC6:7F9D:2A4A (talk) 17:37, 4 August 2024 (UTC)[reply]

The entries in Pascal's triangle are the coefficients of anbn-i in (a+b)n. It's easy to see that (a+b)2=(a2+b2) mod 2, and consequently (a+b)4=(a4+b4) mod 2, (a+b)8=(a8+b8) mod 2, and in general (a+b)n=(an+bn) mod 2 if n is a power of 2. This implies that all the coefficients of anbn-i are 0 mod 2 except when i=0 or i=n. Note, there are more complex formulas for finding anbn-i mod 2 or any other prime for any values of n and i. --RDBury (talk) 18:40, 4 August 2024 (UTC)[reply]
@RDBury: Did you mean aibn-i in the first sentence...? --CiaPan (talk) 09:44, 5 August 2024 (UTC)[reply]
Yes, good catch. RDBury (talk) 13:20, 5 August 2024 (UTC)[reply]
Another way to see it is, we divide   into two equal piles of size  , so that to select k things from   is to select i things from one pile and k-i things from the other. That is, we have the following special case of Vandermonde's identity:
 
In the right-hand side, every term appears twice, except the middle term (if k is even). We thus have
 
We iterate this until k is either odd (and the right hand side is even by induction), or   in which case   or  , which corresponds one of the two outer binomial coefficients. Tito Omburo (talk) 18:51, 4 August 2024 (UTC)[reply]
(OP) I think the first approach more my level, but thanks to both for the replies.2A00:23C6:AA0D:F501:6CF2:A683:CAFC:3D8 (talk) 07:10, 5 August 2024 (UTC)[reply]