LHS =\left({n\atop k}\right)={n!\over k! (n-k)!}
RHS =\left( {n-1\atop k-1}\right)+\left({n-1 \atop k}\right)
={(n-1)!\over (k-1)! (n-1-(k-1))!}+ {(n-1)!\over k! (n-1-k)!}
={(n-1)!\over (k-1)! (n-k)!}+ {(n-1)!\over k! (n-k-1)!}
={k(n-1)!\over k(k-1)! (n-k)!}+ {(n-1)!(n-k)\over k! (n-k)(n-k-1)!}
={k(n-1)!\over k! (n-k)!}+ {(n-1)!(n-k)\over k! (n-k)!}
={k(n-1)! + (n-1)!(n-k)\over k! (n-k)!}
={(n-1)!(k + (n-k))\over k! (n-k)!}
={(n-1)!n\over k! (n-k)!}
={n!\over k! (n-k)!}
={(n-1)!(k + (n-k))\over k! (n-k)!}
={(n-1)!n\over k! (n-k)!}
={n!\over k! (n-k)!}
=LHS
as required.
No comments:
Post a Comment