document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1116837>Question 1116837</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A tray contains 15 chocolate chip cookies and 19 oatmeal raisin cookies. How many ways can a child select 4 cookies that include at least 1 chocolate chip cookie? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #731763 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=731763\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assuming the cookies of each type are indistinguishable…. <br>\r<BR>\n" );
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document.write( "Counting directly the number of possible ways of selecting 4 cookies with at least one or more chocolate chip cookies:  <BR>\n" );
document.write( " 1 choc chip + 3 oatmeal raisin<BR>\n" );
document.write( " 2 choc chips + 2 oatmeal raisin<BR>\n" );
document.write( " 3 choc chips + 1 oatmeal raisin<BR>\n" );
document.write( "  4 choc chips + 0 oatmeal raisin<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+4%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways\r<BR>\n" );
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document.write( "Using stars & bars….<br>\r<BR>\n" );
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document.write( "Again assuming indistinguishable cookies within each type,  this problem maps to a stars and bars problem (with one adjustment) where there are two types (k=2) and we must select 4 of them (n=4). <br> \r<BR>\n" );
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document.write( "For k bars (types) and n stars (number of cookies), the arrangements are:<BR>\n" );
document.write( "C(n+k-1, k-1) = C(n+k-1, n) <br>\r<BR>\n" );
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document.write( "Plugging in n=4, k=2:   C(4+2-1, 1) =  C(5,1) = 5 <br>\r<BR>\n" );
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document.write( "The adjustment…<BR>\n" );
document.write( "The above answer includes the case no chocolate chip cookies are selected, so we must subtract 1  for that case since it is a condition of the problem that at least one of that type is selected.  That leaves us with 5-1 or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+4+%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+4+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways.<BR>\n" );
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document.write( "It may be tempting to use C(15,4)+C(15,3)+C(15,2)+C(15,1) or other counting formulas but those don't apply if the cookies of each type are indistinguishable.\r<BR>\n" );
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