Editing Generating function (section)

====Example: The money-changing problem====

The number of ways to pay {{math|''n'' ≥ 0}} cents in coin denominations of values in the set {1,&nbsp;5,&nbsp;10,&nbsp;25,&nbsp;50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively), where we distinguish instances based upon the total number of each coin but not upon the order in which the coins are presented, is given by the ordinary generating function
<math display="block">\frac{1}{1-z} \frac{1}{1-z^5} \frac{1}{1-z^{10}} \frac{1}{1-z^{25}} \frac{1}{1-z^{50}}\,.</math>
When we also distinguish based upon the order in which the coins are presented (e.g., one penny then one nickel is distinct from one nickel then one penny), the ordinary generating function is
<math display=block>\frac{1}{1-z-z^5-z^{10}-z^{25}-z^{50}}\,.</math>

If we allow the {{mvar|n}} cents to be paid in coins of ''any'' positive integer denomination, we arrive at the  [[partition function (mathematics)|partition function]] ordinary generating function expanded by an infinite [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] product,
<math display="block">\prod_{n = 1}^\infty \left(1 - z^n\right)^{-1}\,.</math>
When the order of the coins matters, the ordinary generating function is
<math display=block>\frac{1}{1-\sum_{n=1}^\infty z^n} = \frac{1-z}{1-2z}\,.</math>