First slide

Operations on Sets

Question

If $n (A) = n$ then $n {(x, y, z); x, y, z \in A$ $x \neq y, y \neq z, z \neq x} =$

Moderate

A

n³
B

$n {(n - 1)}^{2}$
C

$n^{2} (n - 2)$
D

$n^{3} - 3 n^{2} + 2 n$

Solution

There are n choices for the first coordinate, $n - 1$ choices for second coordinate and $n - 2$ choices for the third coordinate, hence
$n ({(x, y, z); x, y, z \in A, x \neq y \neq z}) =$ $n (n - 1) (n - 2) = n^{3} - 3 n^{2} + 2 n$

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