document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=999165>Question 999165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A jar of alphabet tiles contains 10 unique consonant tiles and 5 unique vowel tiles.<BR>\n" );
document.write( "If 5 tiles are picked randomly, the probability that 3 are consonants and 2 are vowels is ?<BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #616865 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=616865\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Problem:<BR>\n" );
document.write( "Given 15 tiles, with 10 unique consonants, and 5 unique vowels.<BR>\n" );
document.write( "Pick 5 randomly (without replacement), find probability of picking 3 consonants and 2 vowels.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "The simplest way is to use the Hypergeometric distribution, with <BR>\n" );
document.write( "N=15 (total number of tiles)<BR>\n" );
document.write( "n=5  (number of tiles picked)<BR>\n" );
document.write( "k=10 (number of successes, initially)<BR>\n" );
document.write( "r=3  (number of successes for which to find),<BR>\n" );
document.write( "finally, <BR>\n" );
document.write( "C(a,b)=number of combinations of choosing b objects out of a<BR>\n" );
document.write( "      = a!/(b!(a-b)!)<BR>\n" );
document.write( "then<BR>\n" );
document.write( "P(r=3)=C(n,r)C(N-n,k-r)/C(N,k)<BR>\n" );
document.write( "=C(5,3)C(15-5,10-3)/C(15,10)<BR>\n" );
document.write( "=10*120/3003<BR>\n" );
document.write( "=400/1001<BR>\n" );
document.write( " <BR>\n" );
document.write( "Another way to approach it would be:<BR>\n" );
document.write( "Let <BR>\n" );
document.write( "C=event of drawing a consonant<BR>\n" );
document.write( "V=event of drawing a vowel<BR>\n" );
document.write( "Then <BR>\n" );
document.write( "P(CCCVV)=(10/15)(9/14)(8/13)(5/12)(4/11)=40/1001<BR>\n" );
document.write( "Since there are C(5,3) ways to arrange 3 consonants and 2 vowels, we multiply<BR>\n" );
document.write( "the above by C(5,3)=5!/(3!2!)=10<BR>\n" );
document.write( "So the final answer is P(r=3)=40/1001*10=400/1001 as before. </SPAN>\n" );
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