​$$\text{If } (a + b + c) = 12, \text{ and } (a^2 + b^2 + c^2) = 50, \text{ find the value of } (a^3 + b^3 + c^3 - 3abc).$$​​

Correct option is A

Given:

​$$If (a + b + c) = 12, and (a^2 + b^2 + c^2) = 50$$​​

Formula Used:

​$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$​​

​$$(a^3 + b^3 + c^3 - 3abc) = (a + b + c)((a^2 + b^2 + c^2) - (ab + bc + ca))$$​​

Solution:

​$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$​​

​$$12^2 = 50 + 2(ab + bc + ca)$$​​

144 = 50 + 2(ab + bc + ca)

2(ab + bc + ca) = 144 - 50 = 94

ab + bc + ca = 94/2 = 47

​$$(a^3 + b^3 + c^3 - 3abc) = (a + b + c)((a^2 + b^2 + c^2) - (ab + bc + ca))$$​​

Substituting the known values:

​$$(a^3 + b^3 + c^3 - 3abc) = 12(50 - 47)\\\ \\(a^3 + b^3 + c^3 - 3abc) = 12 × 3 = 36$$​​

Thus, the value of $$(a^3 + b^3 + c^3 - 3abc)$$​is 36.