document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=9147>Question 9147</A>: <I><SPAN STYLE=\"word-break: break-all;\">  I need help on finding the number of different starting teams if five basketball players are chosen from a team of ten players </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #5171 by </B><A HREF=/tutors/aboutme.mpl?userid=DWL><B>DWL(56)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DWL><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=5171\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a permutation and combination problem.  I'll Give you the answer, but you need to read about these two subjects.\r<BR>\n" );
document.write( "\n" );
document.write( "Permutation Part\r<BR>\n" );
document.write( "\n" );
document.write( "Now we have 10 players and want to make groupings of 5 people. It's harder to list all those permutations. To find the number of five-people permutations that we can make from 10 people without repeated (10_P_5), we'd like to have a formula because there are 30,240 such permutations and we don't want to write them all out!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For five-people permutations, there are 10 possibilities for the first person, 9 for the second, 8 for the third, and 7 for the fourth, and 6 for the last person. We can find the total number of different five-people permutations by multiplying 10 x 9 x 8 x 7 x 6 = 30,240. This is part of a factorial\r<BR>\n" );
document.write( "\n" );
document.write( "To arrive at 10 x 9 x 8 x 7 x 6, we need to divide 10 factorial (10 because there are ten objects) by (10-5) factorial (subtracting from the total number of objects from which we're choosing the number of objects in each permutation). You can see below that we can divide the numerator by 5 x 4 x 3 x 2 x 1:\r<BR>\n" );
document.write( "\n" );
document.write( "10_P_5 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F%2810-5%29 ALIGN=MIDDLE  ALT=\"10%21%2F%2810-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F5%21 ALIGN=MIDDLE  ALT=\"10%21%2F5%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%285%2A4%2A3%2A2%2A1%29 ALIGN=MIDDLE  ALT=\"%2810%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%285%2A4%2A3%2A2%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2A9%2A8%2A7%2A6%29=+30240 ALIGN=MIDDLE  ALT=\"%2810%2A9%2A8%2A7%2A6%29=+30240\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Combination Part\r<BR>\n" );
document.write( "\n" );
document.write( "we have 10 people from which we wish to choose 5 and we want to find the number of combinations of size 5 without repeated people that can be made from the ten people. To calculate 10_C_5, which is 120, we don't want to have to write all the combinations out!\r<BR>\n" );
document.write( "\n" );
document.write( "Since we already know that 10_P_5 = 30,240, we can use this information to find 10_C_5. Let's think about how we got that answer of 30240. We found all the possible combinations of 5 that can be taken from 10 (10_C_5). Then we found all the ways that five people in those groups of size 5 can be arranged: 5 x 4 x 3 x 2 x 1 = 5! = 120. Thus the total number of permutations of size 5 taken from a set of size 10 is equal to 5! times the total number of combinations of size 5 taken from a set of size 10: 10_P_5 = 5! x 10_C_5.\r<BR>\n" );
document.write( "\n" );
document.write( "When we divide both sides of this equation by 5! we see that the total number of combinations of size 5 taken from a set of size 10 is equal to the number of permutations of size 5 taken from a set of size 10 divided by 5!. This makes it possible to write a formula for finding 10_C_5:\r<BR>\n" );
document.write( "\n" );
document.write( "10_C_5 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810_P_5%29%2F5%21 ALIGN=MIDDLE  ALT=\"%2810_P_5%29%2F5%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F5%21%2A5%21 ALIGN=MIDDLE  ALT=\"10%21%2F5%21%2A5%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21%2F5%21%2810-5%21%29 ALIGN=MIDDLE  ALT=\"10%21%2F5%21%2810-5%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%2F%285%2A4%2A3%2A2%2A1%29%285%2A4%2A3%2A2%2A1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2A9%2A8%2A7%2A6%29%2F%285%2A4%2A3%2A2%2A1%29 ALIGN=MIDDLE  ALT=\"%2810%2A9%2A8%2A7%2A6%29%2F%285%2A4%2A3%2A2%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30240%2F120=252 ALIGN=MIDDLE  ALT=\"30240%2F120=252\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "  <BR>\n" );
document.write( "So the answer is:\r<BR>\n" );
document.write( "\n" );
document.write( "You can have 252 starting teams of five from a group of 10 players.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "           </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );