Editing Combinatorial number system (section)

=== Example ===

Suppose one wants to determine the 5-combination at position 72. The successive values of <math>\tbinom n5</math> for ''n''&nbsp;=&nbsp;4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for ''n''&nbsp;=&nbsp;8. Therefore ''c''<sub>5</sub>&nbsp;=&nbsp;8, and the remaining elements form the {{nowrap|4-combination}} at position {{nowrap|72 − 56 {{=}} 16}}. The successive values of <math>\tbinom n4</math> for ''n''&nbsp;=&nbsp;3, 4, 5, ... are 0, 1, 5, 15, 35, ..., of which the largest one not exceeding 16 is 15, for ''n''&nbsp;=&nbsp;6, so ''c''<sub>4</sub>&nbsp;=&nbsp;6. Continuing similarly to search for a 3-combination at position {{nowrap|16 − 15 {{=}} 1}} one finds ''c''<sub>3</sub>&nbsp;=&nbsp;3, which uses up the final unit; this establishes <math>72=\tbinom85+\tbinom64+\tbinom33</math>, and the remaining values ''c''<sub>''i''</sub> will be the maximal ones with <math>\tbinom{c_i}i=0</math>, namely {{nowrap|''c''<sub>''i''</sub> {{=}} ''i'' − 1}}. Thus we have found the 5-combination {{nowrap|{8, 6, 3, 1, 0}}}.