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Consider the expression \(\frac{(A!)}{((B!)^x * (C!)^y * (D!)^z))}\) where A,B,C,D,x,y,z are all positive integers >=1. Is this expression an integer ?
The answer is B This is one of those questions me and a friend of mine made up to test each other (both of us have GMATs coming up). So I have a solution with me, but it is rather unconventional :
It is easy to prove that (1) alone cant be the answer since there is no constraint on x,y,z and one can make these big enough to exceed the numerator. Eg 7! / (2!^40 * 3!^20 * 1!^1)
To show that (2) alone is sufficient : Consider the question "How many permutations are possible of a set of A alphabets, of which x alphabets are each repeated B times, y alphabets each repeated C times and z alphabets each repeated D times within the set ?" The answer to this question is exactly the expression above, and we know that since it is the answer to a counting question, it must be an integer.
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Imagine I ask you the question : What is the number of ways you can arrange A balls, each of different color in a row ? The answer would be \(A!\)
Now I modify that question : What is the number of ways you can arrange A balls, of which B are are blue, C are red, D are green and the rest are of different but unique colors ? The answer would now be \(\frac{A!}{B!C!D!}\)
Now I modify it further : What is the number of ways you can arrange A balls, of which there are x subsets each consisting B balls each such that each subset consists of balls of a different shade of blue, and all other balls not included in these subsets are of unique colors ? The answer would now be \(\frac{A!}{B!^x}\). Notice that A has to be greater than or equal to xB
Finally I modify it a bit more : What is the number of ways you can arrange A balls, of which there are x subsets each consisting B balls each, y subsets of C balls each, z subsets of D balls each, such that each subset consists of a unique color of balls. And the rest of the set of balls are distinct from all other balls ? The answer would now be \(\frac{A!}{B!^xC!^yD!^z}\). Notice that A has to be greater than or equal to xB+yC+zD
Essentially what I am getting at is that the expression shown above is the answer to a combinatorial problem. And since the answer to a combinatorial problem is a "number of ways", such an expression always has to be an integer.
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Version 8.1 of the WordPress for Android app is now available, with some great enhancements to publishing: background media uploading. Adding images to a post or page? Now...
“Keep your head down, and work hard. Don’t attract any attention. You should be grateful to be here.” Why do we keep quiet? Being an immigrant is a constant...
“Keep your head down, and work hard. Don’t attract any attention. You should be grateful to be here.” Why do we keep quiet? Being an immigrant is a constant...
42% (00:46) correct
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