document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=719395>Question 719395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Australian coins are made in 6 denominations:  1&#8373;,  2&#8373;,  5&#8373;,  10&#8373;,  20&#8373;,  50&#8373; and $1.<BR>\n" );
document.write( "Your tasks are to determine all the different ways a person could make from those coins total amounts of:<BR>\n" );
document.write( "7&#8373;<BR>\n" );
document.write( "10&#8373;<BR>\n" );
document.write( "20&#8373; (answer here is >35)<BR>\n" );
document.write( "30&#8373; (here without using any 1&#8373; coins)<BR>\n" );
document.write( "For each of the 4 cases, all the ways must be shown in a logically sequenced list and you must carefully explain in words the logical sequencing “rules” you used so the reader can see that your list is complete.<BR>\n" );
document.write( "P.S. The summation of all the ways across all 4 cases will approach 75. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #441367 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=441367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "To make 7-cents, we can use either 0 or 1 5-cent coin.\r\n" );
document.write( "\r\n" );
document.write( "If we use 0 5-cent coin, we can use from 0 to 3, inclusive, 2-cent coins\r\n" );
document.write( "and the rest in 1-cent coins.  That's 4 ways.\r\n" );
document.write( "\r\n" );
document.write( "If we use 1 5-cent coin, we can get the remaining 2 cents by either 1\r\n" );
document.write( "2-cent coin, or 2 1-cent coins.  That's 2 more ways.\r\n" );
document.write( "\r\n" );
document.write( "Total number of ways for having 7-cents is 4+2 or 6 ways.\r\n" );
document.write( "\r\n" );
document.write( "Here are the 6 ways:\r\n" );
document.write( "\r\n" );
document.write( "&#8373;   5   2   1\r\n" );
document.write( "-------------     \r\n" );
document.write( "1.  0   0   7\r\n" );
document.write( "2.  0   1   5\r\n" );
document.write( "3.  0   2   3\r\n" );
document.write( "4.  0   3   1\r\n" );
document.write( "5.  1   0   2\r\n" );
document.write( "6.  1   1   0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "---------------\r\n" );
document.write( " \r\n" );
document.write( "To make 10-cents, we can use either 0 or 1 10-cent coin.\r\n" );
document.write( "\r\n" );
document.write( "I.   If we use 0 10-cent coin, then we can use 0,1, or 2 5-cent coins:\r\n" );
document.write( "     A.  If we use 0 5-cent coins, then we can use 0 through 5 2-cent coins,\r\n" );
document.write( "         inclusive, 2-cent coins and the rest, if any, in 1-cent coins.  \r\n" );
document.write( "         That's 6 ways.\r\n" );
document.write( "     B.  If we use 1 5-cent coin, then we can use 0 through 2, inclusive,\r\n" );
document.write( "         2-cent coins and the rest, if any, in 1-cent coins.  That's 3 ways.\r\n" );
document.write( "     C.  If we use 2 5-cent coins, that's 10 cents, so that's 1 way.\r\n" );
document.write( "\r\n" );
document.write( "II.  If we use 1 10-cent coin, that's 1 way.\r\n" );
document.write( "\r\n" );
document.write( "Total number of ways for having 10-cents is 6+3+1+1 or 11 ways. \r\n" );
document.write( "\r\n" );
document.write( "Here are the 11 ways:\r\n" );
document.write( "\r\n" );
document.write( " &#8373;  10   5   2   1\r\n" );
document.write( "------------------- \r\n" );
document.write( " 1.  0   0   0  10\r\n" );
document.write( " 2.  0   0   1   8\r\n" );
document.write( " 3.  0   0   2   6\r\n" );
document.write( " 4.  0   0   3   4\r\n" );
document.write( " 5.  0   0   4   2\r\n" );
document.write( " 6.  0   0   5   0\r\n" );
document.write( " 7.  0   1   0   5\r\n" );
document.write( " 8.  0   1   1   3\r\n" );
document.write( " 9.  0   1   2   1\r\n" );
document.write( "10.  0   2   0   0\r\n" );
document.write( "11.  1   0   0   0\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "\r\n" );
document.write( "To make 20-cents, we can use either 0 or 1 20-cent coin.\r\n" );
document.write( "\r\n" );
document.write( "I.   Use 0 20-cent coin, then we can use 0,1,or 2 10-cent coins.\r\n" );
document.write( "     A. Use 0 10-cent coins, then we can use 0,1,2,3, or 4 5-cent coins.   \r\n" );
document.write( "        1. 0 5-cent coins. then we can use 0 through 10, inclusive,\r\n" );
document.write( "           2-cent coins, and the rest, if any, in 1-cent coins.  \r\n" );
document.write( "           That's 11 ways.\r\n" );
document.write( "        2. 1 5-cent coins. then we have 15 cents left to make. Then we \r\n" );
document.write( "           can use 0 through 7, inclusive, 2-cent coins, and the rest\r\n" );
document.write( "           in 1-cent coins.  That's 8 ways.\r\n" );
document.write( "        3. 2 5-cent coins. then we have 10 cents left to make. Then we \r\n" );
document.write( "           can use 0 through 5, inclusive, 2-cent coins, and the rest\r\n" );
document.write( "           in 1-cent coins.  That's 6 ways.\r\n" );
document.write( "        4. 3 5-cent coins. then we have 5 cents left to make. Then we \r\n" );
document.write( "           can use 0 through 2, inclusive, 2-cent coins, and the rest\r\n" );
document.write( "           in 1-cent coins.  That's 3 ways.\r\n" );
document.write( "        5. 4 5-cent coins.  That's 20 cents. That's 1 way.\r\n" );
document.write( "     B. Use 1 10-cent coin.  Then to make the remaining 10-cents, this\r\n" );
document.write( "        is the same as the preceding problem (of making 10 cents) minus\r\n" );
document.write( "        the one case where we used exactly 1 10-cent coin to make the 10\r\n" );
document.write( "        cents.  That's 11-1 or 10 ways.\r\n" );
document.write( "     C. Use 2 10-cent coins.  That's 20 cents.  That's 1 way.\r\n" );
document.write( "II.  1 20-cent coin.  That's 1 way.\r\n" );
document.write( "\r\n" );
document.write( "Total number of ways for having 20-cents is 11+8+6+3+1+10+1+1 or 41 ways.\r\n" );
document.write( "\r\n" );
document.write( "Here are the 42 ways:\r\n" );
document.write( "\r\n" );
document.write( " &#8373;  20  10   5   2   1\r\n" );
document.write( "----------------------\r\n" );
document.write( " 1.  0   0   0   0  20\r\n" );
document.write( " 2.  0   0   0   1  18\r\n" );
document.write( " 3.  0   0   0   2  16\r\n" );
document.write( " 4.  0   0   0   3  14\r\n" );
document.write( " 5.  0   0   0   4  12\r\n" );
document.write( " 6.  0   0   0   5  10\r\n" );
document.write( " 7.  0   0   0   6   8\r\n" );
document.write( " 8.  0   0   0   7   6\r\n" );
document.write( " 9.  0   0   0   8   4\r\n" );
document.write( "10.  0   0   0   9   2\r\n" );
document.write( "11.  0   0   0  10   0\r\n" );
document.write( "12.  0   0   1   0  15\r\n" );
document.write( "13.  0   0   1   1  13\r\n" );
document.write( "14.  0   0   1   2  11\r\n" );
document.write( "15.  0   0   1   3   9\r\n" );
document.write( "16.  0   0   1   4   7\r\n" );
document.write( "17.  0   0   1   5   5\r\n" );
document.write( "18.  0   0   1   6   3\r\n" );
document.write( "19.  0   0   1   7   1\r\n" );
document.write( "20.  0   0   2   0  10\r\n" );
document.write( "21.  0   0   2   1   8\r\n" );
document.write( "22.  0   0   2   2   6\r\n" );
document.write( "23.  0   0   2   3   4\r\n" );
document.write( "24.  0   0   2   4   2\r\n" );
document.write( "25.  0   0   2   5   0\r\n" );
document.write( "26.  0   0   3   0   5\r\n" );
document.write( "27.  0   0   3   1   3\r\n" );
document.write( "28.  0   0   3   2   1\r\n" );
document.write( "29.  0   0   4   0   0\r\n" );
document.write( "30.  0   1   0   0  10\r\n" );
document.write( "31.  0   1   0   1   8\r\n" );
document.write( "32.  0   1   0   2   6\r\n" );
document.write( "33.  0   1   0   3   4\r\n" );
document.write( "34.  0   1   0   4   2\r\n" );
document.write( "35.  0   1   0   5   0\r\n" );
document.write( "36.  0   1   1   0   5\r\n" );
document.write( "37.  0   1   1   1   3\r\n" );
document.write( "38.  0   1   1   2   1\r\n" );
document.write( "39.  0   1   2   0   0\r\n" );
document.write( "40.  0   2   0   0   0\r\n" );
document.write( "41.  1   0   0   0   0\r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "\r\n" );
document.write( "To make 30-cents, we can use either 0 or 1 20-cent coins.  Since we aren't\r\n" );
document.write( "using any 1-cent coins, we can only use an even number of 5-cent coins.\r\n" );
document.write( "\r\n" );
document.write( "I.   If we use 0 20-cent coin, then we can use 0,1,2 or 3 10-cent coins:\r\n" );
document.write( "     A.  If we use 0 10-cent coins, then we can use 0,2,4 or 6 2-cent coins,\r\n" );
document.write( "         inclusive, 2-cent coins and the rest, if any, in 2-cent coins.  \r\n" );
document.write( "         That's 4 ways.\r\n" );
document.write( "     B.  If we use 1 10-cent coin, then we can use 0,2, or 4 5-cent coins and\r\n" );
document.write( "         the rest, if any, in 2-cent coins.  That's 3 ways.\r\n" );
document.write( "     C.  If we use 2 10-cent coins, then we can use 0 or 2 5-cent coins and\r\n" );
document.write( "         the rest, if any, in 2-cent coins.  That's 2 ways.\r\n" );
document.write( "     D.  If we use 3 10-cent coins, that's 30 cents.  That's 1 way.\r\n" );
document.write( "\r\n" );
document.write( "II.  If we use 1 20-cent coin, then we can use 0 or 1 10-cent coin\r\n" );
document.write( "     A. If we use 0 10-cent coins, then we can use 0 or 2 5-cent\r\n" );
document.write( "        coins, and the rest, if any, in 2-cent coins. That's 2 ways.\r\n" );
document.write( "     B. If we use 1 10-cent coin, then that's 30 cents. That's 1 way.\r\n" );
document.write( "\r\n" );
document.write( "Total number of ways for having 30-cents is 4+3+2+1+2+1 or 13 ways. \r\n" );
document.write( "\r\n" );
document.write( "Here are the 13 ways:\r\n" );
document.write( "\r\n" );
document.write( " &#8373;  20  10   5   2 \r\n" );
document.write( "------------------\r\n" );
document.write( " 1.  0   0   0  15   \r\n" );
document.write( " 2.  0   0   2  10   \r\n" );
document.write( " 3.  0   0   4   5   \r\n" );
document.write( " 4.  0   0   6   0   \r\n" );
document.write( " 5.  0   1   0  10   \r\n" );
document.write( " 6.  0   1   2   5   \r\n" );
document.write( " 7.  0   1   4   0   \r\n" );
document.write( " 8.  0   2   0   5   \r\n" );
document.write( " 9.  0   2   2   0   \r\n" );
document.write( "10.  0   3   0   0   \r\n" );
document.write( "11.  1   0   0   5   \r\n" );
document.write( "12.  1   0   2   0   \r\n" );
document.write( "13.  1   1   0   0   \r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );