Let j and n be positive integers, with j≤n. An experiment consists of choosing, at random, a j-tuple of positive integers whose sum is at most n.
((a)) Find the size of the sample space. Hint: Consider n indistinguishable balls placed in a row. Place j markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the n 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 j-tuples whose size we are trying to count.
((b)) Find the probability that the j-tuple selected contains at least one 1.