document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192625>Question 1192625</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different 4-letter permutations can be formed from the letters in the word WESTINGHOUSE? 	 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824523 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824523\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We'll need to break things into multiple cases.\r<BR>\n" );
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document.write( "Here's the table of contents<BR>\n" );
document.write( "Case A) All four letters are unique<BR>\n" );
document.write( "Case B) We have two copies of \"E\" and everything else is unique.<BR>\n" );
document.write( "Case C) We have two copies of \"S\" and everything else is unique.<BR>\n" );
document.write( "Case D) We have two copies of \"E\" and two copies of \"S\"\r<BR>\n" );
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document.write( "Case A) All four letters are unique<BR>\n" );
document.write( "Example word could be: WEST\r<BR>\n" );
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document.write( "In the word \"WESTINGHOUSE\", the unique letters in the order presented are: <BR>\n" );
document.write( "{W, E, S, T, I, N, G, H, O, U}<BR>\n" );
document.write( "There are 10 letters in this list. <BR>\n" );
document.write( "Notice that I didn't write the second copy of \"S\" nor the second copy of \"E\".\r<BR>\n" );
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document.write( "If it might help, then we can write the list of letters out like this<BR>\n" );
document.write( "{W, E, S, T, I, N, G, H, O, U, S, E}<BR>\n" );
document.write( "but cross out the last two letters since they were mentioned earlier<BR>\n" );
document.write( "{W, E, S, T, I, N, G, H, O, U, <s><font color=red>S, E</font></s>}<BR>\n" );
document.write( "so that's how I got to<BR>\n" );
document.write( "{W, E, S, T, I, N, G, H, O, U}\r<BR>\n" );
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document.write( "Since we have n = 10 unique letters to work with, and r = 4 slots to fill, this means we can use the nPr permutation formula. Order matters.\r<BR>\n" );
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document.write( "n P r = (n!)/( (n-r)! )<BR>\n" );
document.write( "10 P 4 = (10!)/( (10-4)! )<BR>\n" );
document.write( "10 P 4 = (10!)/(6!)<BR>\n" );
document.write( "10 P 4 = (10*9*8*7*6!)/(6!)<BR>\n" );
document.write( "10 P 4 = 10*9*8*7<BR>\n" );
document.write( "10 P 4 = 5040\r<BR>\n" );
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document.write( "Or perhaps more intuitively, we start off with 10 and count down by 1 each time until we have four slots to fill. So that helps explain why 10*9*8*7 is buried in the nPr formula when n = 10 and r = 4.\r<BR>\n" );
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document.write( "There are 5040 ways to select four letters such that all of them are unique.\r<BR>\n" );
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document.write( "Let A = 5040 so we can use it later.\r<BR>\n" );
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document.write( "Case B) We have two copies of \"E\" and everything else is unique.<BR>\n" );
document.write( "Example \"word\" could be: WESE\r<BR>\n" );
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document.write( "Let's say we label the slots 1,2,3,4.<BR>\n" );
document.write( "There are 4 choices to place \"E\", and then there are 3 choices left to place the other E. That gives 4*3 = 12, but we divide by 2 since we cannot tell the \"E\"s apart. That leads us to 12/2 = 6 ways to place the \"E\"s.\r<BR>\n" );
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document.write( "Put another way, there are 4 C 2 = 6 ways to place those two \"E\"s. I'm using the nCr combination formula.\r<BR>\n" );
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document.write( "Go back to<BR>\n" );
document.write( "{W, E, S, T, I, N, G, H, O, U, S, E}<BR>\n" );
document.write( "and cross off the duplicate \"S\" and both copies of \"E\"<BR>\n" );
document.write( "{W, <s><font color=red>E</font></s>, S, T, I, N, G, H, O, U, <s><font color=red>S</font></s>, <s><font color=red>E</font></s>}<BR>\n" );
document.write( "we're left with <BR>\n" );
document.write( "{W, S, T, I, N, G, H, O, U}<BR>\n" );
document.write( "which has n = 9 letters in it.\r<BR>\n" );
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document.write( "After placing those \"E\"s, there are r = 2 slots left to fill.<BR>\n" );
document.write( "Meaning we have nPr = 9 P 2 = 9*8 = 72 ways to fill the remaining two slots.\r<BR>\n" );
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document.write( "We found there are 6 ways to place the \"E\"s and 72 ways to select the remaining other unique letters.\r<BR>\n" );
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document.write( "There are 6*72 = 432 ways to carry out case B.\r<BR>\n" );
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document.write( "Let B = 432 so we can use it later.\r<BR>\n" );
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document.write( "Case C)  We have two copies of \"S\" and everything else is unique.<BR>\n" );
document.write( "Example \"word\" could be: WSSE\r<BR>\n" );
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document.write( "The steps are identical to case B.<BR>\n" );
document.write( "This is because \"S\" repeats just as much as \"E\" does.\r<BR>\n" );
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document.write( "This means there are 432 ways to place the two \"S\" letters, and fill in the remaining slots with other unique letters.\r<BR>\n" );
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document.write( "Let C = 432\r<BR>\n" );
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document.write( "Case D) We have two copies of \"E\" and two copies of \"S\"<BR>\n" );
document.write( "Example word: SEES\r<BR>\n" );
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document.write( "There are 6 ways to place the \"E\"s in whatever two slots they land in. Refer to the 4 C 2 = 6 mentioned earlier in case B. The two \"S\" letters will have no choice but to fill the remaining empty slots in exactly one way.\r<BR>\n" );
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document.write( "The order of which you pick doesn't matter, so you could start with the \"S\" letters first, then move onto the \"E\"s later.\r<BR>\n" );
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document.write( "In short, there are 6 ways to carry out case D.\r<BR>\n" );
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document.write( "Let D = 6.\r<BR>\n" );
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document.write( "To wrap everything up, we'll add up the results of each case mentioned earlier.<BR>\n" );
document.write( "Such addition is possible because cases A,B,C,D are mutually exclusive. <BR>\n" );
document.write( "Only one case can happen at a time and there is no overlap.\r<BR>\n" );
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document.write( "A+B+C+D = 5040+432+432+6 = <font color=red>5910</font>\r<BR>\n" );
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document.write( "This can be verified by using computer software to generate all the possible permutations.\r<BR>\n" );
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document.write( "This is a really awesome free calculator that will do just that<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html\">https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html</a><BR>\n" );
document.write( "It not only lists the numeric nCr and nPr values, but it will also list all the actual sequences possible. <BR>\n" );
document.write( "Though I imagine there's a size limit so it cannot list every possible set.<BR>\n" );
document.write( "Luckily, our set isn't too large.\r<BR>\n" );
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document.write( "Type these letters<BR>\n" );
document.write( "W, E, S, T, I, N, G, H, O, U, S, E<BR>\n" );
document.write( "into the box where it mentions \"List Them:\"<BR>\n" );
document.write( "Don't input any curly braces or quotation marks. Each letter must be separated by a comma.<BR>\n" );
document.write( "Make sure to change the \"Types to choose from?\" to 12 and \"Number Chosen?\" to 4. <BR>\n" );
document.write( "Set \"is order important\" to \"yes\", and set \"repetition allowed?\" to \"no\".<BR>\n" );
document.write( "Otherwise, the solver will likely list the wrong thing you're after.<BR>\n" );
document.write( "Check out the screenshot to see what I mean.<BR>\n" );
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document.write( "Click on the \"list\" button at the bottom to have the solver mention <font color=red>5910</font> different four letter permutations after tossing the duplicates.<BR>\n" );
document.write( "Before tossing the duplicates, there are 12 P 4 = 12*11*10*9 = 11,880 entries.\r<BR>\n" );
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document.write( "Answer: <font color=red>5910</font><BR>\n" );
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