If $$a + b + c = 5$$, and $$a^2 + b^2 + c^2 = 33$$, then what is the value of $$a^3 + b^3 + c^3 - 3abc$$?

Given that,
$$a + b + c = 5$$
$$a^2 + b^2 + c^2 = 33$$

From the identity $$a^3 + b^3 + c^3 - 3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$--------(i)

But we know that $$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$$---------(ii)
now substituting the values in the equation(ii)
$$5^2=33+2(ab+bc+ca)$$
$$2(ab+bc+ca)=25-33=-8$$
$$(ab+bc+ca)=-4$$--------(iii)

Now from the equation (i) and (iii)
$$a^3 + b^3 + c^3 - 3abc=(a+b+c)(a^2+b^2+c^2-(ab+bc+ca))$$
Now substituting the values in the above equation
$$a^3 + b^3 + c^3 - 3abc=(5)(33-(-4))$$
$$a^3 + b^3 + c^3 - 3abc=(5)(33+4)$$
$$a^3 + b^3 + c^3 - 3abc=(5)(37)$$
$$a^3 + b^3 + c^3 - 3abc=185$$