document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023733>Question 1023733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 24 is added to a certain number and the sum is divided by 6, the result is 1 more than 13 of the original number. Find the original number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639211 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639211\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "A 4-card poker hand is dealt at random from a standard 52-card deck. <BR>\n" );
document.write( "(a) What is the total number of possible hands?<BR>\n" );
document.write( "(b) What is the total number of possible hands if the hand contains exactly one heart? <BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( " <BR>\n" );
document.write( "One very useful tool is the binomial coefficient, which is in fact the number of possible combinations for r objects taken from n distinct objects:<BR>\n" );
document.write( "C(n,r)=n!/(r!(n-r)!)<BR>\n" );
document.write( " <BR>\n" );
document.write( "Say we have 5 fruits, an apple, an orange, a plum, a pear and a banana.<BR>\n" );
document.write( "The number of different ways we can choose two fruits out of the five is<BR>\n" );
document.write( "C(5,2)   [5 choose 2]<BR>\n" );
document.write( "=5!/(2!3!)<BR>\n" );
document.write( "=120/(2*6)<BR>\n" );
document.write( "=10<BR>\n" );
document.write( " <BR>\n" );
document.write( "For a four-card hand from 52 (distinct) cards deck, the idea is the same.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) possible hands<BR>\n" );
document.write( "C(52,4)<BR>\n" );
document.write( "=52!/(4!48!)<BR>\n" );
document.write( "=52*51*50*49/(1*2*3*4)<BR>\n" );
document.write( "=270725<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) possible hands with only one heart<BR>\n" );
document.write( "We have to have one heart, so we choose 1 heart from the 13 (13 choose 1)<BR>\n" );
document.write( "in C(13,1)=13 ways.<BR>\n" );
document.write( "We choose the remaining 3 cards from the 39 remaining cards in C(39,3) ways.<BR>\n" );
document.write( "C(39,3)=39*38*37/(1*2*3)=9139 ways<BR>\n" );
document.write( "So the number of possible hands is the product of 13 and 9139=118807<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );