Larry King brought up in a 2001 USA Today column that there were 293 ways to make change for one dollar, in currently minted denominations. This is including the one dollar coin that was popular years ago but has since fallen out of favor. Using a half dollar there are 50 combinations and 242 combinations without, getting the 293 grand total.
Warning: Math mumbo-jumbo incoming. To get the answer for any denomination you use trhe coefficient of x^100 in the reciprocal of the following: (1-x)(1-x^5)(1-x^10)(1-x^25)(1-x^50)(1-x^100). That's multiplying everything with every coin denomination (penny, nickel, dime, etc).
$2 bills were allowed and there was no distinguishing between a dollar bill and a dollar coin.
There is some dispute, however, since the dictionary states that "change" is "a number of coins or bills whose total value equals a single larger coin or bill." Therefore, $1 can't be change for $1, skewing the results a bit.
