document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=917652>Question 917652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. The Knick play the Bulls in a seven game series that end when one team has won four games. We record the outcome of a game with a W for a Knick win and an L for a Knick loss.\r<BR>\n" );
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document.write( "How many series would have to be played to be sure that the same outcome happens twice?\r<BR>\n" );
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document.write( "I know there are a total of 840 possible outcomes from a permutation of (7,4), so would it just be 840 divided by 2? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #556757 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=556757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have seven games. There are 2 outcomes for each game.\r<BR>\n" );
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document.write( "There are 2^7 = 128 ways to have a Ws and Ls in a seven character sequence; however, you cannot have something like WWWWWLL because once you get that 4th W, the series is over. So in reality, you the upper limit on Ws is 4. The same is said for the L's as well.\r<BR>\n" );
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document.write( "So here is a list of all the possible outcomes. \r<BR>\n" );
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document.write( "For example, scenario #3 has \"LLLWWL\" under the \"Game Sequence\" column. This means we have the Knicks lose 3 times in a row (L, L, L). The next game has the Knicks winning (W). The Knicks win again in game 5 (W). However, the Knicks lose (L) game 6. Overall, they lose the championship series because the opponent has gained 4 wins before them. This is what I mean by \"Knicks lose\" in the \"Championship Summary\" column.\r<BR>\n" );
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document.write( "The other rows are read in a similar way. Let me know if you have any questions on that.\r<BR>\n" );
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document.write( "There are 70 possible outcomes.\r<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "<TABLE  border = 1>\n" );
document.write( "\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <th>Number</th>\r\n" );
document.write( "    <th>Game Sequence</th>\r\n" );
document.write( "    <th>Championship Summary</th>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >1</TD>\r\n" );
document.write( "    <TD >LLLL</TD>\r\n" );
document.write( "    <TD >4 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >2</TD>\r\n" );
document.write( "    <TD >LLLWL</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >3</TD>\r\n" );
document.write( "    <TD >LLLWWL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >4</TD>\r\n" );
document.write( "    <TD >LLLWWWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >5</TD>\r\n" );
document.write( "    <TD >LLLWWWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >6</TD>\r\n" );
document.write( "    <TD >LLWLL</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >7</TD>\r\n" );
document.write( "    <TD >LLWLWL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >8</TD>\r\n" );
document.write( "    <TD >LLWLWWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >9</TD>\r\n" );
document.write( "    <TD >LLWLWWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >10</TD>\r\n" );
document.write( "    <TD >LLWWLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >11</TD>\r\n" );
document.write( "    <TD >LLWWLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >12</TD>\r\n" );
document.write( "    <TD >LLWWLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >13</TD>\r\n" );
document.write( "    <TD >LLWWWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >14</TD>\r\n" );
document.write( "    <TD >LLWWWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >15</TD>\r\n" );
document.write( "    <TD >LLWWWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >16</TD>\r\n" );
document.write( "    <TD >LWLLL</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >17</TD>\r\n" );
document.write( "    <TD >LWLLWL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >18</TD>\r\n" );
document.write( "    <TD >LWLLWWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >19</TD>\r\n" );
document.write( "    <TD >LWLLWWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >20</TD>\r\n" );
document.write( "    <TD >LWLWLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >21</TD>\r\n" );
document.write( "    <TD >LWLWLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >22</TD>\r\n" );
document.write( "    <TD >LWLWLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >23</TD>\r\n" );
document.write( "    <TD >LWLWWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >24</TD>\r\n" );
document.write( "    <TD >LWLWWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >25</TD>\r\n" );
document.write( "    <TD >LWLWWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >26</TD>\r\n" );
document.write( "    <TD >LWWLLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >27</TD>\r\n" );
document.write( "    <TD >LWWLLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >28</TD>\r\n" );
document.write( "    <TD >LWWLLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >29</TD>\r\n" );
document.write( "    <TD >LWWLWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >30</TD>\r\n" );
document.write( "    <TD >LWWLWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >31</TD>\r\n" );
document.write( "    <TD >LWWLWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >32</TD>\r\n" );
document.write( "    <TD >LWWWLLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >33</TD>\r\n" );
document.write( "    <TD >LWWWLLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >34</TD>\r\n" );
document.write( "    <TD >LWWWLW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >35</TD>\r\n" );
document.write( "    <TD >LWWWW</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >36</TD>\r\n" );
document.write( "    <TD >WLLLL</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >37</TD>\r\n" );
document.write( "    <TD >WLLLWL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >38</TD>\r\n" );
document.write( "    <TD >WLLLWWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >39</TD>\r\n" );
document.write( "    <TD >WLLLWWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >40</TD>\r\n" );
document.write( "    <TD >WLLWLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >41</TD>\r\n" );
document.write( "    <TD >WLLWLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >42</TD>\r\n" );
document.write( "    <TD >WLLWLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >43</TD>\r\n" );
document.write( "    <TD >WLLWWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >44</TD>\r\n" );
document.write( "    <TD >WLLWWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >45</TD>\r\n" );
document.write( "    <TD >WLLWWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >46</TD>\r\n" );
document.write( "    <TD >WLWLLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >47</TD>\r\n" );
document.write( "    <TD >WLWLLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >48</TD>\r\n" );
document.write( "    <TD >WLWLLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >49</TD>\r\n" );
document.write( "    <TD >WLWLWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >50</TD>\r\n" );
document.write( "    <TD >WLWLWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >51</TD>\r\n" );
document.write( "    <TD >WLWLWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >52</TD>\r\n" );
document.write( "    <TD >WLWWLLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >53</TD>\r\n" );
document.write( "    <TD >WLWWLLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >54</TD>\r\n" );
document.write( "    <TD >WLWWLW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >55</TD>\r\n" );
document.write( "    <TD >WLWWW</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >56</TD>\r\n" );
document.write( "    <TD >WWLLLL</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >57</TD>\r\n" );
document.write( "    <TD >WWLLLWL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >58</TD>\r\n" );
document.write( "    <TD >WWLLLWW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >59</TD>\r\n" );
document.write( "    <TD >WWLLWLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >60</TD>\r\n" );
document.write( "    <TD >WWLLWLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >61</TD>\r\n" );
document.write( "    <TD >WWLLWW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >62</TD>\r\n" );
document.write( "    <TD >WWLWLLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >63</TD>\r\n" );
document.write( "    <TD >WWLWLLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >64</TD>\r\n" );
document.write( "    <TD >WWLWLW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >65</TD>\r\n" );
document.write( "    <TD >WWLWW</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >66</TD>\r\n" );
document.write( "    <TD >WWWLLLL</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks lose</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >67</TD>\r\n" );
document.write( "    <TD >WWWLLLW</TD>\r\n" );
document.write( "    <TD >7 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >68</TD>\r\n" );
document.write( "    <TD >WWWLLW</TD>\r\n" );
document.write( "    <TD >6 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >69</TD>\r\n" );
document.write( "    <TD >WWWLW</TD>\r\n" );
document.write( "    <TD >5 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "  <tr>\r\n" );
document.write( "    <TD >70</TD>\r\n" );
document.write( "    <TD >WWWW</TD>\r\n" );
document.write( "    <TD >4 game series, Knicks win</TD>\r\n" );
document.write( "  </tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Again there are 70 possible outcomes. The other 128-70 = 58 outcomes are not possible. This would be something like WWWWWLL (5 W's aren't possible).\r<BR>\n" );
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document.write( "I didn't use a formula. I simply listed out all of the possible outcomes (all 128) using this <a  rel=nofollow HREF=\"http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html\">combinatorics calculator</a>. Then I picked out the outcomes that weren't possible.\r<BR>\n" );
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document.write( "So this means that 71 series would have to be played to make sure the same outcome happens twice. They need to play 70 series to generate all of the outcomes. That 71st game will be a repeat of one of the 70 already listed.\r<BR>\n" );
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