document.write( "Question 1119006: How many ways can you spilt $614 in twenty’s, ten’s and dollar bills? \n" ); document.write( "

Algebra.Com's Answer #734440 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Clearly, using only 20s, 10s, and 1s, the number of 1s to make a total of $614 must be a number with units digit 4. So make an organized list of the different possible numbers of $1 bills; for each of those numbers, find the number of combinations of 20s and 10s that make the remainder of the $614.

\n" ); document.write( "For each such case, that will be easy; using only 20s and 10s, the number of ways to make the remaining amount is just the number of choices you have for the number of 20s.

\n" ); document.write( "For example, if there are 104 $1 bills, the remaining amount is $510. The number of 20s you can have is any whole number for which the sum of the $20 bills is less than or equal to $510. That means 0 to 25 $20 bills, making 26 ways to make the $614 total using 104 $1 bills.

\n" ); document.write( "Then look for a pattern in the numbers of ways for the different cases to find an easy way to determine the total number of ways.

\n" ); document.write( "So...

\r\n" );
document.write( "  # of $1  remaining  # of\r\n" );
document.write( "   bills    amount    ways\r\n" );
document.write( "  -------------------------\r\n" );
document.write( "    4       610       31  (0 to 30)\r\n" );
document.write( "   14       600       31  (0 to 30)\r\n" );
document.write( "   24       590       30  (0 to 29)\r\n" );
document.write( "   34       580       30  (0 to 29)\r\n" );
document.write( "   44       570       29\r\n" );
document.write( "  ...\r\n" );
document.write( "  584        30        2  (0 or 1)\r\n" );
document.write( "  594        20        2  (0 or 1)\r\n" );
document.write( "  604        10        1  (0)\r\n" );
document.write( "  614         0        1  (0)

\n" ); document.write( "The total number of ways is then seen to be

\n" ); document.write( "2(1+2+...+31) = 2((31*32)/2) = 992 \n" ); document.write( "

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