document.write( "Question 892939: A shop has 8 different types of post cards and at least 12 of each type. How many selections of 12 postcards are there if
\n" ); document.write( "a) you can choose any number of each type?
\n" ); document.write( "b) you can choose at most one of type 1 and any number of the other types? \n" ); document.write( "
Algebra.Com's Answer #541065 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
A shop has 8 different types of post cards and at least 12 of each type. How many selections of 12 postcards are there if
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document.write( "Suppose the 8 types are numbered 1,2,...,8\r\n" );
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document.write( "Suppose the 12 cards are in a stack so that the cards of a lower type number\r\n" );
document.write( "are above any cards of a higher type number.\r\n" );
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document.write( "Now let's think of inserting 7 cardboard partitions \r\n" );
document.write( "into the stack of 12 to separate the cards of different types.  \r\n" );
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document.write( "The 1st cardboard partition we insert will have all cards of type 1, if any,\r\n" );
document.write( "just above it and all cards of type 2, if any, just below it.   \r\n" );
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document.write( "The 2nd cardboard partition we insert will have all cards of type 2, if any, \r\n" );
document.write( "just above it and all cards of type 3, if any, just below it.\r\n" );
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document.write( "etc., etc,\r\n" );
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document.write( "The 7th cardboard partition we insert will have all cards of type 7, if any, \r\n" );
document.write( "just above it and all cards of type 8, if any, just below it.\r\n" );
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document.write( "Now you see why we only need 7 partitions to separate the 8 types\r\n" );
document.write( "of cards.\r\n" );
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document.write( "[For instance, in the case all 12 cards are type 1, all 7 cardboard partitions\r\n" );
document.write( "will be together at the bottom of the stack. And for instance, if they are all\r\n" );
document.write( "of type 8, then all 7 partitions will be together at the top of the stack. And\r\n" );
document.write( "for instance, if they are all of type 4, there will be three cardboard\r\n" );
document.write( "partitions on top of the stack and 4 partitions on the bottom of the stack.]\r\n" );
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document.write( "So we now have a stack of 19 things, 12 cards and 7 cardboard partitions.\r\n" );
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document.write( "Now let's imagine that we have a stack of 19 slips of paper.  We choose 7 of\r\n" );
document.write( "those slips of paper and write the word \"partition\" on them, leaving them stuck\r\n" );
document.write( "out a ways from the stack. Then on the slip stuck out nearest the top of the\r\n" );
document.write( "stack we write \r\n" );
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document.write( "\"This slip of paper represents a partition with all type 1 cards, if any, just\r\n" );
document.write( "above it and all type 2 cards, if any, just below it.\"  \r\n" );
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document.write( "We do that with all 7 pieces of paper stuck out.  On the 7th one we write \r\n" );
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document.write( "\"This slip of paper represents a partition with all type 7 cards, if any, just\r\n" );
document.write( "above it and all type 8 cards, if any, just below it.\"   \r\n" );
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document.write( "On all the other pieces of paper between two slips with \"partition\" written on\r\n" );
document.write( "them we will write this: \r\n" );
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document.write( "\"This slip of paper represents a postcard of type (whatever)\".  \r\n" );
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document.write( "There is a 1-to-1 correspondence between the ways we can do this with the slips\r\n" );
document.write( "of paper and the ways we can select the 12 cards.\r\n" );
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document.write( "We can choose those 7 pieces of paper from the 19 to write \"partition\" on in\r\n" );
document.write( "C(19,7) = 50388 ways.   That's the answer.\r\n" );
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document.write( "[Yes, I know, I could have just given you the formula for the number of\r\n" );
document.write( "partitions of n=12 into r=8 terms counting 0's as terms as  C(n+r-1,r-1).\r\n" );
document.write( "But I thought I would show you WHY the formula works!]\r\n" );
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\n" ); document.write( " b) you can choose at most one of type 1 and any number of the other types?
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document.write( "Case 1: We choose none of type 1.\r\n" );
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document.write( "Then we have only 7 types to choose from.  n=12, r=7.  Using the formula\r\n" );
document.write( "above, C(n+r-1,r-1) = C(12+7-1,7-1) = C(18,6) = 18564 ways\r\n" );
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document.write( "Case 2: We choose one of type 1.\r\n" );
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document.write( "After choosing the 1 card of type 1, we have only 11 cards to choose and \r\n" );
document.write( "only 7 types to choose them from.\r\n" );
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document.write( "n=11, r=7.  Using the formula, \r\n" );
document.write( "C(n+r-1,r-1) = C(11+7-1,7-1) = C(17,6) = 12376.\r\n" );
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document.write( "Total for both cases 18564+12376 = 30940. \r\n" );
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document.write( "Edwin
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