The values of the four variables $a$, $b$, $c$, and $d$ are 9, 11, 13, and 15, though not necessarily in that order. What is the number of possible values of the expression ab+bc+cd+da?

michaelcai
Nov 3, 2017

#1**+1 **

I'll give this one a shot.....whether it's correct....Mmmmmm...!!!!!

ab+bc+cd+da =

ab + ad + bc + dc =

a(b + d) + c (b + d) =

(a + c) (b + d)

Note that the only possible values, no matter the arrangements, are

(9 + 11) (13 + 15) = 360

(9 + 13) (11 + 15) = 572

(9+ 15) (13 + 11) = 576

To see this more clearly.....we are choosing any 2 of the 4 numbers to occupy the first set of parentheses without regard to order ....and this means that the second sum is "fixed"

So.....C(4,2) = 6

But we can permute the order of the parentheses' sums in 2 ways.....so 6 / 2 = 3 different values

CPhill
Nov 3, 2017