Two-argument form See also one-argument form Factorial or Gamma function
In its two-argument form, with positive arguments, ! tells you how many different ways there are of selecting the number of items specified on the left from the population of items specified on the right. The order of items in each pair is ignored. So if the population of four consisted of the letters A B C D, the 6 possible combinations of 2 letters would be: AB AC AD BC BD CD. The combination BA would be regarded as the same as AB.
2 ! 4 (Number of unique pairs from a population 6 of 4) 3 ! 20 (Number of groups of three from a 1140 population of 20) 2 ! 6 12 20 (Number of pairs from a population 15 66 190 of 6 12 20 respectively) TABLE1 „ 2 3½¼6 TABLE2 „ 2 3 ½ 3 6 9 12 15 18 TABLE1 ! TABLE2 (TABLE1 is table of group sizes, TABLE2 3 15 84 is table of populations) 495 3003 18564
Other cases, such as negative or non-integer arguments, are also catered for. The various results that can be obtained are:
Left Right Right-Left Result Argument Argument +ve +ve +ve (!RIGHT)÷(!LEFT)×!RIGHT-LEFT +ve +ve -ve 0 +ve -ve -ve (¯1*LEFT)×LEFT!LEFT-RIGHT+1 -ve +ve +ve 0 -ve -ve +ve (¯1*RIGHT-LEFT)×(|RIGHT+1)!(|LEFT+1) -ve -ve -ve 0