Question 453172

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{_{25}C_{5}=]{{{25!/(25-5)!5!}}} Plug in {{{n=25}}} and {{{r=5}}}




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{25!/20!5!}}}  Subtract {{{25-5}}} to get 20



Expand 25!
*[Tex \LARGE \textrm{_{25}C_{5}=]{{{(25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/20!5!}}}



Expand 20!
*[Tex \LARGE \textrm{_{25}C_{5}=]{{{(25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)/(20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1)5!}}}




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{(25*24*23*22*21*cross(20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1))/(cross(20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1))5!}}}  Cancel




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{(25*24*23*22*21)/5!}}}  Simplify



Expand 5!
*[Tex \LARGE \textrm{_{25}C_{5}=]{{{(25*24*23*22*21)/(5*4*3*2*1)}}}




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{6375600/(5*4*3*2*1)}}}  Multiply 25*24*23*22*21 to get 6,375,600




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{6375600/120}}} Multiply 5*4*3*2*1 to get 120




*[Tex \LARGE \textrm{_{25}C_{5}=]{{{53130}}} Now divide




So 25 choose 5 (where order doesn't matter) yields 53,130 unique combinations



So there are 53,130 different ways to form a group of 5 people.