The evaluation of (2-a)³+(2-b)³+(2-c)³-3(2-a)(2-b)(2-c) is mathematically given as
a^3+3 a b c-b^3-c^3+6 a^2-6 a b-6 a c+6 b^2-6 b c+6 c^2
This is further explained below.
(2-a)^3+(2-b)^3+(2-c)^3-3(2-a)(2-b)(2-c)
Use the Binomial Theorem.
2^3+3 * 2^2(-a)+3*2(-a)^2+(-a)^3 +(2-b)^3+(2-c)^3 -3(2-a)(2-b)(2-c)
8-12 a+6 a^2-a^3+(2-b)^3+(2-c)^3 -3(2-a)(2-b)(2-c)
Use the Binomial Theorem.
8-12 a+6 a^2-a^3+2^3+3 * 2^2(-b)+3 * 2(-b)^2+(-b)^3+(2-c)^3-3(2-a)(2-b)(2-c)
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3 +(2-c)^3-3(2-a)(2-b)(2-c)
Use the Binomial Theorem.
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3+2^3+3 \cdot 2^2(-c)+3 \cdot 2(-c)^2+(-c)^3-3(2-a)(2-b)(2-c)
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3 +8-12 c+6 c^2-c^3-3(2-a)(2-b)(2-c)
Apply the distributive property.
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3+8-12 c+6 c^2-c^3+(-3 * 2-3(-a))(2-b)(2-c)
Multiply -3 by 2 .
&8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3+8-12 c+6 c^2-c^3 +(-6-3(-a))(2-b)(2-c)
Multiply -1 by -3.
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3 +8-12 c+6 c^2-c^3 +(-6+3 a)(2-b)(2-c)
Expand (-6+3 a)(2-b) using the FOIL Method.
Combine the opposite terms in
8-12 a+6 a^2-a^3+8-12 b+6 b^2-b^3+8-12 c+6 c^2-c^3-24+12 c+12 b-6 b c+12 a-6 a c-6 a b+3 abc
8+6 a^2-a^3+8+6 b^2-b^3+8+6 c^2-c^3-24-6 b c-6 a c-6 a b+3 abc
Simplify by adding numbers.
6 a^2-a^3+6 b^2-b^3+24+6 c^2-c^3-24-6 b c-6 a c-6 a b+3 abc
Combine the opposite terms in
6 a^2-a^3+6 b^2-b^3+24+6 c^2-c^3-24-6 b c-6 a c-6 a b+3 a b c
6 a^2-a^3+6 b^2-b^3+6 c^2-c^3-6 b c -6 ac-6 ab+3 abc
Re odrder
a^3+3 a b c-b^3-c^3+6 a^2-6 a b-6 a c+6 b^2-6 b c+6 c^2
Read more about distributive property.
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