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Let n be a positive integer.

(a) There are \(n^2\) ordered pairs  of positive integers, where \(1 \le a,b \le n.\)  Using a counting argument, show that this number is also equal to

\(n + 2 \binom{n}{2}\)

(b) There are \(n^3\) ordered triples  of positive integers, where \(1 \le a,b,c \le n.\)  Using a counting argument, show that this number is also equal to

\(n + 3n(n - 1) + 6 \binom{n}{3}.\)

 Feb 12, 2023
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Here is the answer to part (b);

 

Let's call (a,b,c) a smiley face if b is less than a and b is less than c, because when we plot the graph, we get a happy face!  And if b is greater than a and b is greater than c, that's a frowny face, because we get a frowny face when we turn a smiely face up-side-down.

 

There are other kinds of faces like smirks (like a is less than b and b is less than c) and neutral faces (like when a is equal to b and b is equal to c).  If the face is neutral, then a equals b and b equals c, so when we choose a, b, and c are also chosen, and there are n choices for a, so there are n neutral faces.

 

Now we count the number of smirks.  There are n ways to choose a, and there are n - 1 ways to choose b.  We also multiply by 3, because the value that we chose for a could have also been the value of b, or the value of c.  So there are 3n(n - 1) smirks.

 

Now we count the number of smiley faces.  There are n ways to choose a, then n - 1 ways to choose b, then n - 2 ways to choose c.  So there are n(n - 1)(n - 2) = 3C(n,3) smiley faces.  By symmetry, there are 3C(n,3) frowny faces.

Therefore, the total number of faces is n^3 = n + 3n(n - 1) + 6C(n,3).

 Feb 12, 2023

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