Revision as of 01:47, 15 June 2024 by Admin (Created page with "Prove that <math display="block"> \sum_{j,k} \biggl(\frac 1{3^n}\frac{n!}{j!k!(n-j-k)!}\biggr) = 1\ , </math> where the sum extends over all non-negative <math>j</math> and <math>k</math> such that <math>j + k \le n</math>. '' Hint'': Count how many ways one can place <math>n</math> labelled balls in 3 labelled urns.")
ABy Admin
Jun 15'24
Exercise
Prove that
[[math]]
\sum_{j,k}
\biggl(\frac 1{3^n}\frac{n!}{j!k!(n-j-k)!}\biggr) = 1\ ,
[[/math]]
where the sum extends over all non-negative [math]j[/math] and [math]k[/math] such that [math]j + k \le n[/math]. Hint: Count how many ways one can place [math]n[/math] labelled balls in 3 labelled urns.