Question 251519
*[Tex \LARGE \textrm{_{n}C_{r}=]{{{n!/(n-r)!r!}}} Start with the given formula




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{5!/(5-2)!2!}}} Plug in {{{n=5}}} and {{{r=2}}}




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{5!/3!2!}}}  Subtract {{{5-2}}} to get 3



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



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




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




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{(5*4)/2!}}}  Simplify



Expand 2!
*[Tex \LARGE \textrm{_{5}C_{2}=]{{{(5*4)/(2*1)}}}




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{20/(2*1)}}}  Multiply 5*4 to get 20




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{20/2}}} Multiply 2*1 to get 2




*[Tex \LARGE \textrm{_{5}C_{2}=]{{{10}}} Now divide




So 5 choose 2 (where order doesn't matter) yields 10 unique combinations


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




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{5!/(5-1)!1!}}} Plug in {{{n=5}}} and {{{r=1}}}




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{5!/4!1!}}}  Subtract {{{5-1}}} to get 4



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



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




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




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{(5)/1!}}}  Simplify



Expand 1!
*[Tex \LARGE \textrm{_{5}C_{1}=]{{{(5)/(1)}}}




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{5/(1)}}}  Multiply 5 to get 5




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{5/1}}} Multiply 1 to get 1




*[Tex \LARGE \textrm{_{5}C_{1}=]{{{5}}} Now divide




So 5 choose 1 (where order doesn't matter) yields 5 unique combinations




Since *[Tex \LARGE \textrm{_{5}C_{2}=10] and *[Tex \LARGE \textrm{_{5}C_{1}=5], this means *[Tex \LARGE \textrm{_{5}C_{2} } + \textrm{_{5}C_{1} } = 10 + 5 = 15]