If x=b+c, y=c+a, z=a+b, then what is (x+y+z)^{3}-24xyz equal to?
- A. a^{3}+b^{3}+c^{3}
- B. 2(a^{3}+b^{3}+c^{3})
- C. 8(a^{3}+b^{3}+c^{3}) ✓
- D. None of the above
Correct Answer: C. 8(a^{3}+b^{3}+c^{3})
Explanation
Substitute arbitrary values a=1, b=1, c=1. Then x=2, y=2, z=2. The expression evaluates to (2+2+2)^3 - 24(2)(2)(2) = 216 - 192 = 24. Testing the options, only 8(a^3+b^3+c^3) = 8(1+1+1) = 24 satisfies the condition.
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