Numbers Expressed as Sums of Binomial Coefficients/Examples
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Examples of Numbers Expressed as Sums of Binomial Coefficients
Number base $3$
When $n = 3$ we have:
\(\ds 0\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0\) | $(012)$ | |||||||||
\(\ds 1\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 1\) | $(013)$ | |||||||||
\(\ds 2\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 1\) | $(023)$ | |||||||||
\(\ds 3\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 1\) | $(123)$ | |||||||||
\(\ds 4\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 4\) | $(014)$ | |||||||||
\(\ds 5\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 4\) | $(024)$ | |||||||||
\(\ds 6\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 4\) | $(124)$ | |||||||||
\(\ds 7\) | \(=\) | \(\, \ds \binom 0 1 + \binom 3 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 3 + 4\) | $(034)$ | |||||||||
\(\ds 8\) | \(=\) | \(\, \ds \binom 1 1 + \binom 3 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 3 + 4\) | $(134)$ | |||||||||
\(\ds 9\) | \(=\) | \(\, \ds \binom 2 1 + \binom 3 2 + \binom 4 3 \, \) | \(\, \ds = \, \) | \(\ds 2 + 3 + 4\) | $(234)$ | |||||||||
\(\ds 10\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 10\) | $(015)$ | |||||||||
\(\ds 11\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 10\) | $(025)$ | |||||||||
\(\ds 12\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 10\) | $(125)$ | |||||||||
\(\ds 13\) | \(=\) | \(\, \ds \binom 0 1 + \binom 3 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 3 + 10\) | $(035)$ | |||||||||
\(\ds 14\) | \(=\) | \(\, \ds \binom 1 1 + \binom 3 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 3 + 10\) | $(135)$ | |||||||||
\(\ds 15\) | \(=\) | \(\, \ds \binom 2 1 + \binom 3 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 2 + 3 + 10\) | $(235)$ | |||||||||
\(\ds 16\) | \(=\) | \(\, \ds \binom 0 1 + \binom 4 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 6 + 10\) | $(045)$ | |||||||||
\(\ds 17\) | \(=\) | \(\, \ds \binom 1 1 + \binom 4 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 1 + 6 + 10\) | $(145)$ | |||||||||
\(\ds 18\) | \(=\) | \(\, \ds \binom 2 1 + \binom 4 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 2 + 6 + 10\) | $(245)$ | |||||||||
\(\ds 19\) | \(=\) | \(\, \ds \binom 3 1 + \binom 4 2 + \binom 5 3 \, \) | \(\, \ds = \, \) | \(\ds 3 + 6 + 10\) | $(345)$ | |||||||||
\(\ds 20\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 6 3 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 20\) | $(016)$ |
Number base $4$
When $n = 4$ we have:
\(\ds 0\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 3 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 0\) | $(0123)$ | |||||||||
\(\ds 1\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 1\) | $(0124)$ | |||||||||
\(\ds 2\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 1 + 1\) | $(0134)$ | |||||||||
\(\ds 3\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 1 + 1\) | $(0234)$ | |||||||||
\(\ds 4\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 1 + 1\) | $(1234)$ | |||||||||
\(\ds 5\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 5\) | $(0125)$ | |||||||||
\(\ds 6\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 1 + 5\) | $(0135)$ | |||||||||
\(\ds 7\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 1 + 5\) | $(0235)$ | |||||||||
\(\ds 8\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 1 + 5\) | $(1235)$ | |||||||||
\(\ds 9\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 4 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 4 + 5\) | $(0145)$ | |||||||||
\(\ds 10\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 4 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 4 + 5\) | $(0245)$ |