Numbers Expressed as Sums of Binomial Coefficients/Examples/n = 3

From ProofWiki
Jump to navigation Jump to search

Examples of Numbers Expressed as Sums of Binomial Coefficients

When $n = 3$ we have:

\(\ds 0\) \(=\) \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 \, \) \(\, \ds = \, \) \(\ds 0 + 0 + 0\) $(012)$
\(\ds 1\) \(=\) \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 \, \) \(\, \ds = \, \) \(\ds 0 + 0 + 1\) $(013)$
\(\ds 2\) \(=\) \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 \, \) \(\, \ds = \, \) \(\ds 0 + 1 + 1\) $(023)$
\(\ds 3\) \(=\) \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 \, \) \(\, \ds = \, \) \(\ds 1 + 1 + 1\) $(123)$
\(\ds 4\) \(=\) \(\, \ds \binom 0 1 + \binom 1 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 0 + 0 + 4\) $(014)$
\(\ds 5\) \(=\) \(\, \ds \binom 0 1 + \binom 2 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 0 + 1 + 4\) $(024)$
\(\ds 6\) \(=\) \(\, \ds \binom 1 1 + \binom 2 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 1 + 1 + 4\) $(124)$
\(\ds 7\) \(=\) \(\, \ds \binom 0 1 + \binom 3 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 0 + 3 + 4\) $(034)$
\(\ds 8\) \(=\) \(\, \ds \binom 1 1 + \binom 3 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 1 + 3 + 4\) $(134)$
\(\ds 9\) \(=\) \(\, \ds \binom 2 1 + \binom 3 2 + \binom 4 3 \, \) \(\, \ds = \, \) \(\ds 2 + 3 + 4\) $(234)$
\(\ds 10\) \(=\) \(\, \ds \binom 0 1 + \binom 1 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 0 + 0 + 10\) $(015)$
\(\ds 11\) \(=\) \(\, \ds \binom 0 1 + \binom 2 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 0 + 1 + 10\) $(025)$
\(\ds 12\) \(=\) \(\, \ds \binom 1 1 + \binom 2 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 1 + 1 + 10\) $(125)$
\(\ds 13\) \(=\) \(\, \ds \binom 0 1 + \binom 3 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 0 + 3 + 10\) $(035)$
\(\ds 14\) \(=\) \(\, \ds \binom 1 1 + \binom 3 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 1 + 3 + 10\) $(135)$
\(\ds 15\) \(=\) \(\, \ds \binom 2 1 + \binom 3 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 2 + 3 + 10\) $(235)$
\(\ds 16\) \(=\) \(\, \ds \binom 0 1 + \binom 4 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 0 + 6 + 10\) $(045)$
\(\ds 17\) \(=\) \(\, \ds \binom 1 1 + \binom 4 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 1 + 6 + 10\) $(145)$
\(\ds 18\) \(=\) \(\, \ds \binom 2 1 + \binom 4 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 2 + 6 + 10\) $(245)$
\(\ds 19\) \(=\) \(\, \ds \binom 3 1 + \binom 4 2 + \binom 5 3 \, \) \(\, \ds = \, \) \(\ds 3 + 6 + 10\) $(345)$
\(\ds 20\) \(=\) \(\, \ds \binom 0 1 + \binom 1 2 + \binom 6 3 \, \) \(\, \ds = \, \) \(\ds 0 + 0 + 20\) $(016)$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Numbers_Expressed_as_Sums_of_Binomial_Coefficients/Examples/n_%3D_3&oldid=358381"