document.write( "Question 1203284: A staff car park at a school has 13 parking spaces in a row.
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\n" ); document.write( "a. In how many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
\n" ); document.write( "b. How many different arrangements are there if the 4 empty spaces are next to each other?
\n" ); document.write( "c. If the parking is random, find the probability that there will not be 4 empty spaces next to each other. \n" ); document.write( "
Algebra.Com's Answer #838722 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "A staff car park at a school has 13 parking spaces in a row.
\n" ); document.write( "There are 9 cars to be parked.
\n" ); document.write( "a. In how many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
\n" ); document.write( "b. How many different arrangements are there if the 4 empty spaces are next to each other?
\n" ); document.write( "c. If the parking is random, find the probability that there will not be 4 empty spaces next to each other.
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\n" ); document.write( "\n" ); document.write( "        In my solution, I consider the cars as different / distinguishable.
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document.write( "(a)  We can represent each placement/arrangement of the cars in a parking\r\n" );
document.write( "     as a word of the length 13 consisting of 9 different letters and the 10th symbol,\r\n" );
document.write( "     which is a blank symbol.\r\n" );
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document.write( "    Then the problem is reduced to this question:\r\n" );
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document.write( "        how many words of the length 13 are there, written with 9+1 = 10\r\n" );
document.write( "        different symbols, such that one of these 10 symbols is repeating 4 times?\r\n" );
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document.write( "    It is a standard model/problem of combinatorics for arranging sets with repeating elements.\r\n" );
document.write( "    Its solution is well known. The formula is\r\n" );
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document.write( "        the number of arrangements is \"13%21%2F4%21\" = \"%281%2A2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10%2A11%2A12%2A13%29%2F%281%2A2%2A3%2A4%29\" = 259,459,200.   ANSWER\r\n" );
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document.write( "(b)  Obviously, there are (9-1) + 1 + 1 = 10 way to place the block of 4 blank symbols \r\n" );
document.write( "     in the row of 13 positions, where 9 positions are occupied by 9 letters.    \r\n" );
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document.write( "     In the remaining 9 positions, 9 cars can be placed in 9! ways;\r\n" );
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document.write( "     so, there are 10*9! = 10! = 10*9*8*7*6*5*4*3*2*1 = 3,628,800 different arrangements of this kind.\r\n" );
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document.write( "(c)  To answer (c), we should take the ratio of the number  of 10! = 3,628,800  from (a) and\r\n" );
document.write( "     the number of 259,459,200 from (b) and then to take the complement to this ratio\r\n" );
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document.write( "         P = 1 - \"3628800%2F259459200\" = 0.9860  (rounded).    ANSWER\r\n" );
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