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Question:

If x + y + z = 10, x² + y² + z² = 30, then the value of x³ + y³ + z³ - 3xyz is ________.

Explanation:

Given, x + y + z = 10
Squaring both sides, we get
⇒ x² + y² + z² + 2xy + 2yz + 2zx = 100
⇒ 30 + 2xy + 2yz + 2zx = 100
⇒ 2xy + 2yz + 2zx = 70
⇒ xy + yz + zx = 35

We know, x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

⇒ x³ + y³ + z³ - 3xyz = (10)(30 - 35)

⇒ x³ + y³ + z³ - 3xyz = (10)(-5)

⇒ x³ + y³ + z³ - 3xyz = -50

Hence, option (c).