Question 167636
Since order does not matter (the balls can be arranged in any order), 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 combination formula




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{10!/(10-6)!6!}}} Plug in {{{n=10}}} and {{{r=6}}}




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{10!/4!6!}}}  Subtract {{{10-6}}} to get 4



Expand 10!
*[Tex \LARGE \textrm{_{10}C_{6}=]{{{(10*9*8*7*6*5*4*3*2*1)/4!6!}}}



Expand 4!
*[Tex \LARGE \textrm{_{10}C_{6}=]{{{(10*9*8*7*6*5*4*3*2*1)/(4*3*2*1)6!}}}




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{(10*9*8*7*6*5*cross(4*3*2*1))/(cross(4*3*2*1))6!}}}  Cancel




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{(10*9*8*7*6*5)/6!}}}  Simplify



Expand 6!
*[Tex \LARGE \textrm{_{10}C_{6}=]{{{(10*9*8*7*6*5)/(6*5*4*3*2*1)}}}




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{151200/(6*5*4*3*2*1)}}}  Multiply 10*9*8*7*6*5 to get 151,200




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{151200/720}}} Multiply 6*5*4*3*2*1 to get 720




*[Tex \LARGE \textrm{_{10}C_{6}=]{{{210}}} Divide




So 10 choose 6 (where order doesn't matter) yields 210 unique combinations




So there are 210 possible ways to pick 6 numbers from a pool of 10 numbers.