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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 $ab+bc+cd+da$?

 

Edit: I got 24, but I want to comfirm

michaelcai  Nov 7, 2017
edited by michaelcai  Nov 7, 2017
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 #1
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It should be 4! = 24 values.

Guest Nov 7, 2017
 #2
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The answer was 3.

michaelcai  Nov 7, 2017
 #3
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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?

 

Let a = 9
Let b = 11
Let c = 13
Let d = 15


\(\small{ \begin{array}{|r|c|r|r|r|r|rcl|} \hline & \text{All permutations} & & & & & \\ & \text{of a,b,c,d} & A & B & C & D & AB+BC+CD+DA &=& (B+D)(A+C) \\ \hline 1. & abcd & 9 & 11 & 13 & 15 & (11+15)(9+13) = 26*22 &=& 572 \\ 2. & abdc & 9 & 11 & 15 & 13 & (11+13)(9+15) = 24*24 &=& \qquad 576 \\ 3. & acbd & 9 & 13 & 11 & 15 & (13+15)(9+11) = 28*20 &=& \qquad \qquad 560 \\ 4. & acdb & 9 & 13 & 15 & 11 & (13+11)(9+15) = 24*24 &=& \qquad 576 \\ 5. & adcb & 9 & 15 & 13 & 11 & (15+11)(9+13) = 26*22 &=& 572 \\ 6. & adbc & 9 & 15 & 11 & 13 & (15+13)(9+11) = 28*20 &=& \qquad \qquad 560 \\ 7. & bacd & 11 & 9 & 13 & 15 & (9+15)(11+13) = 24*24 &=& \qquad 576 \\ 8. & badc & 11 & 9 & 15 & 13 & (9+13)(11+15) = 22*26 &=& 572 \\ 9. & bcad & 11 & 13 & 9 & 15 & (13+15)(11+9) = 28*20 &=& \qquad \qquad 560 \\ 10. & bcda & 11 & 13 & 15 & 9 & (13+9)(11+15) = 22*26 &=& 572 \\ 11. & bdca & 11 & 15 & 13 & 9 & (15+9)(11+13) = 24*24 &=& \qquad 576 \\ 12. & bdac & 11 & 15 & 9 & 13 & (15+13)(11+9) = 28*20 &=& \qquad \qquad 560 \\ 13. & cbad & 13 & 11 & 9 & 15 & (11+15)(13+9) = 26*22 &=& 572 \\ 14. & cbda & 13 & 11 & 15 & 9 & (11+9)(13+15) = 20*28 &=& \qquad \qquad 560 \\ 15. & cabd & 13 & 9 & 11 & 15 & (9+15)(13+11) = 24*24 &=& \qquad 576 \\ 16. & cadb & 13 & 9 & 15 & 11 & (9+11)(13+15) = 20*28 &=& \qquad \qquad 560 \\ 17. & cdab & 13 & 15 & 9 & 11 & (15+11)(13+9) = 26*22 &=& 572 \\ 18. & cdba & 13 & 15 & 11 & 9 & (15+9)(13+11) = 24*24 &=& \qquad 576 \\ 19. & dbca & 15 & 11 & 13 & 9 & (11+9)(15+13) = 20*28 &=& \qquad \qquad 560 \\ 20. & dbac & 15 & 11 & 9 & 13 & (11+13)(15+9) = 24*24 &=& \qquad 576 \\ 21. & dcba & 15 & 13 & 11 & 9 & (13+9)(15+11) = 22*26 &=& 572 \\ 22. & dcab & 15 & 13 & 9 & 11 & (13+11)(15+9) = 24*24 &=& \qquad 576 \\ 23. & dacb & 15 & 9 & 13 & 11 & (9+11)(15+13) = 20*28 &=& \qquad \qquad 560 \\ 24. & dabc & 15 & 9 & 11 & 13 & (9+13)(15+11) = 22*26 &=& 572 \\ \hline \end{array} }\)

 

The number of possible values of the expression ab+bc+cd+da=(b+d)(a+c) is 3

The values are 560, 572, and 576

 

laugh

heureka  Nov 7, 2017
edited by heureka  Nov 7, 2017

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