Question : If $a+b+c=7$ and $a^3+b^3+c^3-3abc=175$, then what is the value of $(a b+b c+c a)$?
Option 1: 8
Option 2: 9
Option 3: 7
Option 4: 6
Correct Answer: 8
Solution :
The given equations are:
$a + b + c = 7$
$a^3 + b^3 + c^3 - 3abc = 175$
From the identity,
$a^3 + b^3 + c^3 - 3abc = (a + b + c)[( a + b + c)^2-3(ab + bc + ca)]$
$⇒175 = 7[( 7)^2-3(ab + bc + ca)]$
$⇒25 = 49-3(ab + bc + ca)$
$⇒24 = 3(ab + bc + ca)$
$⇒(ab + bc + ca)=8$
Hence, the correct answer is 8.
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