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Let n be a positive integer.

(a) There are n^2 ordered pairs (a, b) 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 (a, b, c) of positive integers, where  \(1 \le a, b, c \le n\)  Using a counting argument, show that this number is also equal to
\(n + 3n(n - 1) + 6 \binom{n}{3}\)

 Apr 15, 2023
 #1
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(a) Just expand both sides.

 

(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).

 Apr 15, 2023
 #2
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Other facial expressions include neutral expressions and smirks (like an is less than b and b is less than c) (like when a is equal to b and b is equal to c). When an is picked, b and c are also chosen if the face is neutral since an equals b and c equals b when the face is neutral. Since there are n possible options for a, there are n neutral faces.

Guest Apr 16, 2023

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