[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]

A manufactured lot of brass turnbuckles has [math]S[/math] items of

which [math]D[/math] are defective. A sample of [math]s[/math] items is drawn without replacement. Let [math]X[/math] be a random variable that gives the number of defective items in the sample. Let [math]p(d) = P(X = d)[/math].

• Show that
  [[math]] p(d) = \frac{{D \choose d} {{S - D} \choose {s - d}}}{{S \choose s}}\ . [[/math]]
  Thus, X is hypergeometric.
• Prove the following identity, known as Euler's formula:
  [[math]] \sum_{d = 0}^{\min(D,s)}{ D \choose d} {{S - D} \choose {s - d}} = {S \choose s}\ . [[/math]]