Editing Pascal's triangle (section)

== Combinations ==
A second useful application of Pascal's triangle is in the calculation of [[combination]]s. The number of combinations of <math>n</math> items taken <math>k</math> at a time, i.e. the number of subsets of <math>k</math> elements from among <math>n</math> elements, can be found by the equation

:<math> \mathbf{C}(n, k) = \mathbf{C}_{k}^{n}= {{}_{n}C_{k}} = {n \choose k} = \frac{n!}{k!(n-k)!}</math>.

This is equal to entry <math>k</math> in row <math>n</math> of Pascal's triangle. Rather than performing the multiplicative calculation, one can simply look up the appropriate entry in the triangle (constructed by additions). For example, suppose 3 workers need to be hired from among 7 candidates; then the number of possible hiring choices is 7 choose 3, the entry 3 in row 7 of the above table (taking into consideration the first row is the 0th row), which is <math> \tbinom{7}{3}=35 </math>.<ref>{{Cite web |url=http://5010.mathed.usu.edu/Fall2018/HWheeler/probability.html |access-date=2023-06-01 |website=5010.mathed.usu.edu|title=Pascal's Triangle in Probability}}</ref>