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)![]() ![]() You can put this solution on YOUR website! you need:\r \n" ); document.write( " \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 \n" ); document.write( "5 out of 8 seniors\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "c(n,x) is equal to n! / (x! * (n-x)!\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "for example:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "c(8,5) is equal to 8! / (5! * 3!).\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "this is equal to 8 * 7 * 6 * 5! / (5! * 3!)\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "suppose 3 freshmen and 2 sophomores and you want a committee of 2 freshmen and 1 sophomore.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "the calculation would be c(3,2) * (c(2,1).\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "this would be equal to 3 * 2 = 6 possibilities.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "assume the freshmen are a,b,c. \n" ); document.write( "assume the sophomores are 1,2.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "the possible combinations are:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "ab, ac, bc for the freshmen \n" ); document.write( "1, 2 for the sophomores.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "the possible teams are therefore ab1, ac1, bc1, ab2, ac2, bc2.\r \n" ); document.write( " \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 \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "if you make it too simple, you might not see the complexities inherent in the larger set.\r \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "in this case, it is reasonably certain that it does.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |