document.write( "Question 162243: the student council is ordering pizza for their next meeting. there are 20 council members, 7 of whom are vegetarian. A commiittee of 3 will order 6 pizzas from a pizza shop that has a special price for large pizzas with up to three toppings. the shop offers 10 different toppings.
\n" ); document.write( "a) how many different pizza committes can the council choose if there must be at least 1 vegetarian and 1 non vegetarian on the committe?
\n" ); document.write( "b) in how many ways could the committee choose up to 3 toppings for a pizza?
\n" ); document.write( "c) The committee decide to order each topping exactly once and to have at least 1 topping on each pizza. describe the different cases possible?
\n" ); document.write( "d) for one of the cases determine the number of ways of choosing and distributing the 10 toppings \n" ); document.write( "
Algebra.Com's Answer #119567 by Edwin McCravy(20056)\"\" \"About 
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document.write( "The student council is ordering pizza for their next meeting.\r\n" );
document.write( "There are 20 council members, 7 of whom are vegetarian. A \r\n" );
document.write( "committee of 3 will order 6 pizzas from a pizza shop that has \r\n" );
document.write( "a special price for large pizzas with up to three toppings. \r\n" );
document.write( "The shop offers 10 different toppings.\r\n" );
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document.write( "a) How many different pizza committes can the council choose \r\n" );
document.write( "if there must be at least 1 vegetarian and 1 non vegetarian \r\n" );
document.write( "on the committe?\r\n" );
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document.write( "There are 7 vegetarians and 13 non-vegetarians.\r\n" );
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document.write( "The committee either must consist of 1 vegetarian and 2 non-vegetarians or\r\n" );
document.write( "2 vegetarians and 1 non vegetarian.\r\n" );
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document.write( "The number of committees consisting of 1 vegetarian and 2 non-vegetarians\r\n" );
document.write( "is (7C1)(13C2)=7*78=546\r\n" );
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document.write( "The number of committees consisting of 2 vegetarians and 1 non-vegetarian\r\n" );
document.write( "is  (7C2)(13C1)=21*13=273\r\n" );
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document.write( "Therefore total number of acceptable committes is 546 + 273 = 819 \r\n" );
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document.write( "b) in how many ways could the committee choose up to 3 toppings for a pizza?\r\n" );
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document.write( "That says for 'a' pizza, that is, just ONE of the pizzas, not all 6 of them.\r\n" );
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document.write( "If they choose exactly 1 topping, that \"10 choose 1\" or \"10C1=10\"\r\n" );
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document.write( "If they choose exactly 2 toppings, that \"10 choose 2\" or \"10C2=45\"\r\n" );
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document.write( "If they choose exactly 3 toppings, that \"10 choose 3\" or \"10C3=120\"\r\n" );
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document.write( "That gives a total of \"10%2B45%2B120=175\" ways to order just one pizza.\r\n" );
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document.write( "c) The committee decide to order each topping exactly once and to have\r\n" );
document.write( "at least 1 topping on each pizza. describe the different cases possible?\r\n" );
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document.write( "It doesn't say calculate, but just \"describe\" them.  So we can have three\r\n" );
document.write( "cases:\r\n" );
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document.write( "Case 1:\r\n" );
document.write( "2 pizzas with exactly 1 topping each and 4 pizzas with exactly 2 toppings\r\n" );
document.write( "each. \r\n" );
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document.write( "Case 2:\r\n" );
document.write( "3 pizzas with exactly 1 topping each, 2 pizzas with exactly 2 toppings each,\r\n" );
document.write( "and 1 pizza with exactly 3 toppings each.\r\n" );
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document.write( "Case 3:\r\n" );
document.write( "4 pizza with exactly 1 topping each and 2 pizzas with exactly 3 toppings\r\n" );
document.write( "each. \r\n" );
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document.write( "d) for one of the cases determine the number of ways of choosing and\r\n" );
document.write( "distributing the 10 toppings.\r\n" );
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document.write( "It just says for ONE of the cases, so I suppose we can pick whichever\r\n" );
document.write( "case we wish.\r\n" );
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document.write( "Case 1:\r\n" );
document.write( "2 pizzas with exactly 1 topping each and 4 pizzas with exactly 2 toppings\r\n" );
document.write( "each.\r\n" );
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document.write( "That's difficult to avoid counting the same things more than once!\r\n" );
document.write( "But we can do it, but I won't explain it entirely:\r\n" );
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document.write( "The division by 4! was necessary to avoid counting the same things more\r\n" );
document.write( "than once. \r\n" );
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document.write( "Case 2:\r\n" );
document.write( "3 pizzas with exactly 1 topping each, 2 pizzas with exactly 2 toppings \r\n" );
document.write( "each, and 1 pizza with exactly 3 toppings each.\r\n" );
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document.write( "That's also difficult to avoid counting the same things more than once!\r\n" );
document.write( "But we can do it, but I won't explain it entirely:\r\n" );
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document.write( "\"+%28%2810C3%29%2A%288C2%29%286C2%29%29%2F2%21=%28120%2A28%2A15%29%2F2=25200\"\r\n" );
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document.write( "The division by 2! was necessary to avoid counting the same things more\r\n" );
document.write( "than once.\r\n" );
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document.write( "Case 3:\r\n" );
document.write( "4 pizza with exactly 1 topping each and 2 pizzas with exactly 3 toppings \r\n" );
document.write( "each.\r\n" );
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document.write( "That's not too difficult to avoid counting the same thing twice.  So I'll\r\n" );
document.write( "pick that one to explain entirely:  \r\n" );
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document.write( "First we choose a pizza with 3 toppings.  That's 10C3.  That leaves 7\r\n" );
document.write( "toppings for the other pizza with three toppings. That's 7C3. That would\r\n" );
document.write( "be (10C3)(7C3)=120*35=4200.  However this counts ever case twice because:\r\n" );
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document.write( "Take the case where we chose pepperoni, mushrooms and olives for the\r\n" );
document.write( "first pizza, and onions, anchovies, and sausage for the second pizza.\r\n" );
document.write( "That would be the same as if we had chosen onions, anchovies, and sausage\r\n" );
document.write( "for the first pizza and pepperoni, mushrooms and olives for the second \r\n" );
document.write( "pizza.\r\n" );
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document.write( "Therefore we must divide the 4200 by 2 to get the correct count.\r\n" );
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document.write( "4200÷2 = 2100.\r\n" );
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document.write( "So there are 2100 ways to get the number of ways we could have case 3.\r\n" );
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document.write( "We don't have to consider the 4 pizzas with 1 topping each because these\r\n" );
document.write( "will just be 4C4 or 1 way for the remaining 4 toppings to go on the\r\n" );
document.write( "remaining 4 pizzas. \r\n" );
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document.write( "Edwin

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