Question 187873
In this case, order does NOT matter since the candidates have no rank over one another (ie one isn't president or secretary).



Since order does not matter, we must use the <a href=http://www.mathwords.com/c/combination_formula.htm>combination formula</a>:



*[Tex \LARGE \textrm{_{n}C_{r}=]{{{n!/(n-r)!r!}}} Start with the given formula




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{19!/(19-7)!7!}}} Plug in {{{n=19}}} and {{{r=7}}}




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{19!/12!7!}}}  Subtract {{{19-7}}} to get 12



Expand 19!
*[Tex \LARGE \textrm{_{19}C_{7}=]{{{(19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/12!7!}}}



Expand 12!
*[Tex \LARGE \textrm{_{19}C_{7}=]{{{(19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(12*11*10*9*8*7*6*5*4*3*2*1)7!}}}




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{(19*18*17*16*15*14*13*cross(12*11*10*9*8*7*6*5*4*3*2*1))/(cross(12*11*10*9*8*7*6*5*4*3*2*1))7!}}}  Cancel




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{(19*18*17*16*15*14*13)/7!}}}  Simplify. Note: you forgot to divide by 7! (you just divided by 7)



Expand 7!
*[Tex \LARGE \textrm{_{19}C_{7}=]{{{(19*18*17*16*15*14*13)/(7*6*5*4*3*2*1)}}}




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{253955520/(7*6*5*4*3*2*1)}}}  Multiply 19*18*17*16*15*14*13 to get 253,955,520




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{253955520/5040}}} Multiply 7*6*5*4*3*2*1 to get 5,040




*[Tex \LARGE \textrm{_{19}C_{7}=]{{{50388}}} Now divide




So 19 choose 7 (where order doesn't matter) yields 50,388 unique combinations