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Combinations

A second useful application of Pascal's triangle is in the calculation of combinations. The number of combinations of n {\displaystyle n} items taken k {\displaystyle k} at a time, i.e. the number of subsets of k {\displaystyle k} elements from among n {\displaystyle n} elements, can be found by the equation

This is equal to entry k {\displaystyle k} in row of n {\displaystyle n} 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, which is (7 3) = 35 {\displaystyle {\tbinom {7}{3}}=35}.