Numbers Expressed as Sums of Binomial Coefficients/Examples/n = 4
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Examples of Numbers Expressed as Sums of Binomial Coefficients
When $n = 4$ we have:
| \(\ds 0\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 3 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 0\) | $(0123)$ | |||||||||
| \(\ds 1\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 1\) | $(0124)$ | |||||||||
| \(\ds 2\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 1 + 1\) | $(0134)$ | |||||||||
| \(\ds 3\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 1 + 1\) | $(0234)$ | |||||||||
| \(\ds 4\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 + \binom 4 4 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 1 + 1\) | $(1234)$ | |||||||||
| \(\ds 5\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 2 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 0 + 5\) | $(0125)$ | |||||||||
| \(\ds 6\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 1 + 5\) | $(0135)$ | |||||||||
| \(\ds 7\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 1 + 5\) | $(0235)$ | |||||||||
| \(\ds 8\) | \(=\) | \(\, \ds \binom 1 1 + \binom 2 2 + \binom 3 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 1 + 1 + 1 + 5\) | $(1235)$ | |||||||||
| \(\ds 9\) | \(=\) | \(\, \ds \binom 0 1 + \binom 1 2 + \binom 4 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 0 + 4 + 5\) | $(0145)$ | |||||||||
| \(\ds 10\) | \(=\) | \(\, \ds \binom 0 1 + \binom 2 2 + \binom 4 3 + \binom 5 4 \, \) | \(\, \ds = \, \) | \(\ds 0 + 1 + 4 + 5\) | $(0245)$ |
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $56$