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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
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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

 

 

cool cool cool

CPhill  Nov 3, 2017
edited by CPhill  Nov 3, 2017
edited by CPhill  Nov 3, 2017

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