Answer
$$(1+a)(1+b)\cdot(1+c)-1(1+c-1)+1-1+b=abc+ab+ac+bc+a+2b+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+a)(1+b)\cdot(1+c)-1(1+c-1)+1-1+b& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1+a)(1+b)\cdot(1+c)-1\cdot1c+b \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1+a)(1+c+b+bc)-c+b \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}abc+ab+ac+bc+a+b+c+1-c+b \xlongequal{ } \\[1 em] & \xlongequal{ }abc+ab+ac+bc+a+b+ \cancel{c}+1 -\cancel{c}+b \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}abc+ab+ac+bc+a+2b+1\end{aligned} $$ | |
| ① | Combine like terms: $$ \, \color{blue}{ \cancel{1}} \,+c \, \color{blue}{ -\cancel{1}} \, = c $$Combine like terms: $$ \, \color{blue}{ \cancel{1}} \, \, \color{blue}{ -\cancel{1}} \,+b = b $$ |
| ② | Multiply each term of $ \left( \color{blue}{1+b}\right) $ by each term in $ \left( 1+c\right) $. $$ \left( \color{blue}{1+b}\right) \cdot \left( 1+c\right) = 1+c+b+bc $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1+a}\right) $ by each term in $ \left( 1+c+b+bc\right) $. $$ \left( \color{blue}{1+a}\right) \cdot \left( 1+c+b+bc\right) = 1+c+b+bc+a+ac+ab+abc $$ |
| ④ | Combine like terms: $$ 1+c+b+bc+a+ac+ab+abc = abc+ab+ac+bc+a+b+c+1 $$ |
| ⑤ | Combine like terms: $$ abc+ab+ac+bc+a+ \color{blue}{b} + \, \color{red}{ \cancel{c}} \,+1 \, \color{red}{ -\cancel{c}} \,+ \color{blue}{b} = abc+ab+ac+bc+a+ \color{blue}{2b} +1 $$ |
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