+619 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?
+130071 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
