document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203284>Question 1203284</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A staff car park at a school has 13 parking spaces in a row.<BR>\n" );
document.write( "There are 9 cars to be parked.<BR>\n" );
document.write( "a. In how many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?<BR>\n" );
document.write( "b. How many different arrangements are there if the 4 empty spaces are next to each other?<BR>\n" );
document.write( "c. If the parking is random, find the probability that there will not be 4 empty spaces next to each other. </SPAN>\n" );
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document.write( "(a) One way to think of the problem is with 13 letters in a row, with 9 \"F\" representing the parking spaces that are full and 4 \"E\" representing the spaces that are empty.  Then by a well-known counting principle the number of possible arrangements is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2813%21%29%2F%28%289%21%29%284%21%29%29=715 ALIGN=MIDDLE  ALT=\"%2813%21%29%2F%28%289%21%29%284%21%29%29=715\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: 715<br><BR>\n" );
document.write( "Note another way to look at the problem is simply choosing 9 of the 13 spaces to be the ones that get used.  Then the calculation is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2813%2C9%29=%2813%21%29%2F%28%289%21%29%284%21%29%29=715 ALIGN=MIDDLE  ALT=\"C%2813%2C9%29=%2813%21%29%2F%28%289%21%29%284%21%29%29=715\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Of course the answer (and the calculation that was made) is the same.<br><BR>\n" );
document.write( "(b) In the case where the 4 empty spaces are together, think of the parking spaces as being numbered 1 to 13; then the first of the four empty spaces can be space number 1, number 2, ..., or number 10.  So there are 10 arrangements in which the four empty spaces are together.<br><BR>\n" );
document.write( "ANSWER: 10<br><BR>\n" );
document.write( "(c) There are 715 total arrangements, of which 10 have the four empty spaces together, so 705 arrangements have the empty spaces not all together.  So the probability that the four empty spaces will not be together is 705/715.<br><BR>\n" );
document.write( "ANSWER: 705/715 = 141/143<br><BR>\n" );
document.write( "Convert to decimal or percent if desired/required.<br><BR>\n" );
document.write( " </SPAN>\n" );
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