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
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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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