document.write( "Question 890996: In how many ways we can arrange the word 'ENGINEERING' so that all N's come together and no E's come together \n" ); document.write( "
Algebra.Com's Answer #539447 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"%22%22-%22%22\"\"%22%22-%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "The number of ways the N's can come together\r\n" );
document.write( "\r\n" );
document.write( "The letters of ENGINEERING arranged in alphabetical order is\r\n" );
document.write( "\r\n" );
document.write( "E,E,E,G,G,I,I,N,N,N,R\r\n" );
document.write( "\r\n" );
document.write( "The number of ways the three N's can come together is the number of\r\n" );
document.write( "distinguishable permutations of these 9 things, where the (NNN) is\r\n" );
document.write( "considered as a single \"thing\".\r\n" );
document.write( "\r\n" );
document.write( "E,E,E,G,G,I,I,(NNN),R\r\n" );
document.write( "\r\n" );
document.write( "Since there are 3 indistinguishable E's, 2 indistinguishable G's,\r\n" );
document.write( "and 2 indistinguishable I's, the number is:\r\n" );
document.write( "\r\n" );
document.write( "\"9%21%2F%283%212%212%21%29=15120\"\r\n" );
document.write( "\r\n" );
document.write( "From this 15120 we must subtract the number of ways 2 or 3 E's \r\n" );
document.write( "can come together.\r\n" );
document.write( "\r\n" );
document.write( "First we will calculate the number of distinguishable arrangements of\r\n" );
document.write( "these 7 things:\r\n" );
document.write( "\r\n" );
document.write( "(EE),G,G,I,I,(NNN),R  where one E is missing, and then we'll calculate\r\n" );
document.write( "how many ways we can insert the third E into each one of those.  Here\r\n" );
document.write( "the (EE) and the (NNN) are each considered as just one \"thing\".\r\n" );
document.write( "\r\n" );
document.write( "Since there are 2 indistinguishable G's, and 2 indistinguishable I's, \r\n" );
document.write( "the number is:\r\n" );
document.write( "\r\n" );
document.write( "\"7%21%2F%282%212%21%29\" = 1260\r\n" );
document.write( "\r\n" );
document.write( "An example would be GG(EE)II(NNN)R. Let's put a space before and after \r\n" );
document.write( "each letter or \"thing\" to indicate feasible places to insert the third\r\n" );
document.write( "E. We'll number the spaces below each space, like this: \r\n" );
document.write( "\r\n" );
document.write( "  _G_G_(EE)_I_R_(NNN)_I_\r\n" );
document.write( "  1 2 3    4 5 6     7 8\r\n" );
document.write( "\r\n" );
document.write( "We might think at first that we could insert the 3rd E into any of the \r\n" );
document.write( "8 spaces.  However we would not be able to distinguish between placing \r\n" );
document.write( "the third E in positions 3 or 4 above.  That is, we could not tell the\r\n" );
document.write( "difference between the result of inserting the third E just before, or\r\n" );
document.write( "just after, the (EE).  So there are 1 less than 8, or only 7 places that\r\n" );
document.write( "we can insert the third E and have a distinguishable permutation.\r\n" );
document.write( "\r\n" );
document.write( "So we multiply 1260 by the 7 ways to insert the third E.  So 1260×7 = \r\n" );
document.write( "8820.  That is the number which we must subtract from the 15120.\r\n" );
document.write( "\r\n" );
document.write( "Final answer = 15120 - 8820 = 6300 distinguishable permutations.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );