document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202441>Question 1202441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mrs. Green likes to serve two different kinds of vegetables with dinner.  She has carrots, peas, okra, and green beans in her refrigerator .  How many different sets of two vegetables can be serve?  Tell wether this situation is a combination or permutation. </SPAN>\n" );
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document.write( "c = carrots<BR>\n" );
document.write( "p = peas<BR>\n" );
document.write( "o = okra<BR>\n" );
document.write( "g = green beans\r<BR>\n" );
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document.write( "Let's make a table with 4 rows and 4 columns. Along the top and left side will be the letters mentioned<BR>\n" );
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document.write( "<tr><TD ></TD><TD >c</TD><TD >p</TD><TD >o</TD><TD >g</TD></tr><tr><TD >c</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >p</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >o</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >g</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "Assuming Mrs Green cannot select the same veggie twice, we cross off the items along the northwest (or southeast) diagonal.<BR>\n" );
document.write( "For example, the upper left corner has \"carrots\" chosen twice.<BR>\n" );
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document.write( "<tr><TD ></TD><TD >c</TD><TD >p</TD><TD >o</TD><TD >g</TD></tr><tr><TD >c</TD><TD >X</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >p</TD><TD ></TD><TD >X</TD><TD ></TD><TD ></TD></tr><tr><TD >o</TD><TD ></TD><TD ></TD><TD >X</TD><TD ></TD></tr><tr><TD >g</TD><TD ></TD><TD ></TD><TD ></TD><TD >X</TD></tr>\n" );
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document.write( "Let's also cross off items below the diagonal. This is because the table has symmetry along the diagonal. An item below the diagonal will have a mirrored copy above the diagonal<BR>\n" );
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document.write( "<tr><TD ></TD><TD >c</TD><TD >p</TD><TD >o</TD><TD >g</TD></tr><tr><TD >c</TD><TD >X</TD><TD ></TD><TD ></TD><TD ></TD></tr><tr><TD >p</TD><TD >X</TD><TD >X</TD><TD ></TD><TD ></TD></tr><tr><TD >o</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD ></TD></tr><tr><TD >g</TD><TD >X</TD><TD >X</TD><TD >X</TD><TD >X</TD></tr>\n" );
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document.write( "There are 6 cells that haven't been crossed off, which is the final answer.<BR>\n" );
document.write( "For instance, the left-most cell of the top row represents the combo \"carrots and peas\" in either order.\r<BR>\n" );
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document.write( "We started with a 4*4 = 16 cell table<BR>\n" );
document.write( "Then subtracted off the four diagonal elements: 16-4 = 12<BR>\n" );
document.write( "Cut that result in half (because we crossed out the lower half below the diagonal) to get 12/2 = 6<BR>\n" );
document.write( "This division by two operation is to avoid double-counting.\r<BR>\n" );
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document.write( "The general formula is n*(n-1)/2 where n is a positive integer.<BR>\n" );
document.write( "In this case we have n = 4 veggies to get n*(n-1)/2 = 4*(4-1)/2 = 6 combinations. Order doesn't matter. <BR>\n" );
document.write( "A combo like \"carrots and peas\" is the same as \"peas and carrots\".\r<BR>\n" );
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document.write( "Notes: <ul><li>An alternative pathway is to use the nCr combination formula with n = 4 and r = 2.</li><li>Another path is to use Pascal's Triangle. Look at the row that starts with \"1,4,...\" and count over 3 spots to the right because the index starts at r = 0.</li></ul></font> </SPAN>\n" );
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