document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116173>Question 116173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A restaurant offered salads with 4 types of dressings and 7 different toppings.  How many different types of salads could be offered? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84509 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=84509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> That depends.  Can you have more than one type of dressing and/or more than one type of topping?  If not, the answer is trivial:  There are four ways to pick the dressing and seven ways to pick the topping, 7 X 4 = 28 -- assuming that you must pick 1 dressing and 1 topping.  If you were allowed to either not have any dressing or topping or to just have a plain salad, then you would have 5 dressing choices (one of them being no dressing at all) and 8 topping choices, so you would have 5 X 8 = 40 possibilities.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Other situations are more complex to calculate.  For example, if you are allowed two dressings and three toppings, you would first have to decide whether you could also have less than two dressings and three toppings, and whether the order in which the dressings and toppings went on the salad makes a difference or not.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's set some rules and do a calculation so you can see how it is done and perhaps extend the method to more general cases.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's say the rules are:<BR>\n" );
document.write( "4 types of dressing<BR>\n" );
document.write( "7 types of topping<BR>\n" );
document.write( "You may have either 0, 1, or 2 dressings, but you may not duplicate (i.e. if you pick two dressings, they have to be different)<BR>\n" );
document.write( "You may have either 0, 1, 2, or 3 toppings, but you may not duplicate<BR>\n" );
document.write( "The dressing goes on first, but the order in which multiple dressings or toppings go on the salad makes no difference.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is only one way to select no dressing at all, plus<BR>\n" );
document.write( "There are 4 ways to select just one dressing, plus<BR>\n" );
document.write( "if you select 2 dressings, there are 4 ways to pick the first one times 3 ways to pick the second one divided by 2 because for every possible pair of dressings, there are two ways for one to be first and the other second, and we said order of application doesn't matter:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+4+%2B+6=+11 ALIGN=MIDDLE  ALT=\"1+%2B+4+%2B+6=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to pick your dressing.\r<BR>\n" );
document.write( "\n" );
document.write( "There is only one way to select no topping at all, plus<BR>\n" );
document.write( "There are 7 ways to select just one topping, plus<BR>\n" );
document.write( "if you select 2 toppings, there are 7 ways to pick the first one times 6 ways to pick the second one divided by 2 because order doesn't matter, plus<BR>\n" );
document.write( "if you select 3 toppings, there are 7 ways to pick the first one times 6 ways to pick the second one times 5 ways to pick the third one divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2A3=6 ALIGN=MIDDLE  ALT=\"1%2A2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > because order doesn't matter, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B7%2B21%2B35=64 ALIGN=MIDDLE  ALT=\"1%2B7%2B21%2B35=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to pick the topping.\r<BR>\n" );
document.write( "\n" );
document.write( "And finally,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2A64=704 ALIGN=MIDDLE  ALT=\"11%2A64=704\"   DISABLED_style= \"max-width:90%; height:auto;\" > different types of salad.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You see why you have to be specific when you ask a question?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope this helps,<BR>\n" );
document.write( "John </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );