document.write( "Question 1024507: A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 freshmen, 7 sophomores, 7 juniors, and 8 seniors are eligible to be on the committee, in how many ways can the committee be chosen? \n" ); document.write( "
Algebra.Com's Answer #640043 by Theo(13342)\"\" \"About 
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you need:\r
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\n" ); document.write( "\n" ); document.write( "2 out of 6 freshmen
\n" ); document.write( "3 out of 7 sophomores
\n" ); document.write( "4 out of 7 juniors
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\n" ); document.write( "\n" ); document.write( "the number of ways you can get 2 out of 6 freshmen is c(6,2) = 15
\n" ); document.write( "the number of ways you can get 3 out of 7 sophomores is c(7,3) = 35
\n" ); document.write( "the number of ways you can get 4 out of 7 juniors is c(7,4) = 35
\n" ); document.write( "the number of ways you can get 5 out of 8 seniors is c(8,5) = 56\r
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\n" ); document.write( "\n" ); document.write( "the total number of ways you can get 2 freshmen and 3 sophomores and 4 juniors and 5 seniors is therefore 15 * 35 * 35 * 56 = 1,029,000.\r
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\n" ); document.write( "\n" ); document.write( "c(n,x) is equal to n! / (x! * (n-x)!\r
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\n" ); document.write( "\n" ); document.write( "for example:\r
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\n" ); document.write( "\n" ); document.write( "c(8,5) is equal to 8! / (5! * 3!).\r
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\n" ); document.write( "\n" ); document.write( "this is equal to 8 * 7 * 6 * 5! / (5! * 3!)\r
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\n" ); document.write( "\n" ); document.write( "the 5! on the top and the 5! on the bottom cancel out and you are left with (8 * 7 * 6) / (3 * 2 * 1).\r
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\n" ); document.write( "\n" ); document.write( "the 6 on the top and the (3 * 2) on the bottom cancel out and you are left with (8 * 7) / 1 which is equal to 56.\r
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\n" ); document.write( "\n" ); document.write( "to see how this works, make the problem much smallerr so you can detail out the individual possibilities.\r
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\n" ); document.write( "\n" ); document.write( "suppose 3 freshmen and 2 sophomores and you want a committee of 2 freshmen and 1 sophomore.\r
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\n" ); document.write( "\n" ); document.write( "the calculation would be c(3,2) * (c(2,1).\r
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\n" ); document.write( "\n" ); document.write( "this would be equal to 3 * 2 = 6 possibilities.\r
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\n" ); document.write( "\n" ); document.write( "assume the freshmen are a,b,c.
\n" ); document.write( "assume the sophomores are 1,2.\r
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\n" ); document.write( "\n" ); document.write( "the possible combinations are:\r
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\n" ); document.write( "\n" ); document.write( "ab, ac, bc for the freshmen
\n" ); document.write( "1, 2 for the sophomores.\r
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\n" ); document.write( "\n" ); document.write( "the possible teams are therefore ab1, ac1, bc1, ab2, ac2, bc2.\r
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\n" ); document.write( "\n" ); document.write( "the formula looks good for the smaller set, so you can reasonably apply it to the larger set.\r
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\n" ); document.write( "\n" ); document.write( "you do have to make sure that the smaller set and the larger set follow the same rules.\r
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\n" ); document.write( "\n" ); document.write( "if you make it too simple, you might not see the complexities inherent in the larger set.\r
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\n" ); document.write( "\n" ); document.write( "for practical purposes, this type of analysis does work as long as you are careful that the models work both for the simpler situation and the more complex situaiton.\r
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\n" ); document.write( "\n" ); document.write( "in this case, it is reasonably certain that it does.\r
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