[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]j[/math] and [math]n[/math] be positive integers, with [math]j \le n[/math]. An

experiment consists of choosing, at random, a [math]j[/math]-tuple of positive integers whose sum is at most [math]n[/math].

• Find the size of the sample space. Hint: Consider [math]n[/math] indistinguishable balls placed in a row. Place [math]j[/math] markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the [math]n[/math] markers to be placed at the end of the row of balls.) Show that there is a 1-1 correspondence between the set of possible positions for the markers and the set of [math]j[/math]-tuples whose size we are trying to count.
• Find the probability that the [math]j[/math]-tuple selected contains at least one 1.