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In Chapter 2 Section 2.1.4 of An Introduction to Statistical Learning (James et al, 2013), the assertion is made that "...if there are p variables in our data set, then p(p-1)/2 distinct scatter plots can be made...".

Would you please clarify how this is derived?

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  • $\begingroup$ You have p variables to choose for the x-axis and then p-1 for the y-axis. This gives you p(p-1) plots but the order really doesn't matter. So you divide by 2 to get the number of distinct pairs. Note that you have included variable 1 (say) for x-axis and variable 2 for the y-axis and also variable 2 for the x-axis and variable 1 for the y-axis. That is why you have to divide by 2. $\endgroup$ Jul 30, 2017 at 11:46

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Assuming you don't count a plot of $X_3$ vs $X_6$ as distinct from a plot of $X_6$ vs $X_3$ and further assuming you don't care to plot a variable vs itself, then you want the number of distinct pairs $i,j$ for $i<j$ and $i$ and $j$ integers between $1$ and $p$ exclusive.

There's $p \times p$ pairs $(i,j)$. We remove the $i=j$ cases, which removes $p$ of those, leaving $p \times (p-1)$.

diagram of pairs 1 ≤ i < j ≤ p

We then take the half that have $i<j$ (the other half have $i>j$ but they're the same set of plots with axes interchanged). This leaves $\frac12 p\times (p-1)$

Alternatively you could just look at it as the number of ways of choosing two distinct variables out of $p$, without regard to order, which is ${p \choose 2}=p(p-1)/2$.

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