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\begin{array}{c}\phantom{\times99}777\\\underline{\times\phantom{99}123}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times99}777\\\underline{\times\phantom{99}123}\\\phantom{\times9}2331\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 777 with 3. Write the result 2331 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times99}777\\\underline{\times\phantom{99}123}\\\phantom{\times9}2331\\\phantom{\times}1554\phantom{9}\\\end{array}
Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 777 with 2. Write the result 1554 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times99}777\\\underline{\times\phantom{99}123}\\\phantom{\times9}2331\\\phantom{\times}1554\phantom{9}\\\underline{\phantom{\times}777\phantom{99}}\\\end{array}
Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 777 with 1. Write the result 777 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times99}777\\\underline{\times\phantom{99}123}\\\phantom{\times9}2331\\\phantom{\times}1554\phantom{9}\\\underline{\phantom{\times}777\phantom{99}}\\\phantom{\times}95571\end{array}
Now add the intermediate results to get final answer.