Total students = 5 + 5 + 3 = 13

Numbers of students majoring in Computer Science = 5

Numbers of students not majoring in Computer Science = 13 - 5 = 8

Number of groups with at most 2 students majoring in Computer Science = Number of groups with 0 students majoring in Computer Science + Number of groups with 1 students majoring in Computer Science + Number of groups with 2 students majoring in Computer Science

\(\displaystyle={\left({{C}_{{{0}}}^{{{5}}}}\cdot{{C}_{{{3}}}^{{{8}}}}\right)}+{\left({{C}_{{{1}}}^{{{5}}}}\cdot{{C}_{{{2}}}^{{{8}}}}\right)}+{\left({{C}_{{{2}}}^{{{5}}}}\cdot{{C}_{{{1}}}^{{{8}}}}\right)}\)

\(\displaystyle={\left({1}\cdot{56}\right)}+{\left({5}\cdot{28}\right)}+{\left({10}\cdot{8}\right)}\)

\(\displaystyle={276}\)

Numbers of students majoring in Computer Science = 5

Numbers of students not majoring in Computer Science = 13 - 5 = 8

Number of groups with at most 2 students majoring in Computer Science = Number of groups with 0 students majoring in Computer Science + Number of groups with 1 students majoring in Computer Science + Number of groups with 2 students majoring in Computer Science

\(\displaystyle={\left({{C}_{{{0}}}^{{{5}}}}\cdot{{C}_{{{3}}}^{{{8}}}}\right)}+{\left({{C}_{{{1}}}^{{{5}}}}\cdot{{C}_{{{2}}}^{{{8}}}}\right)}+{\left({{C}_{{{2}}}^{{{5}}}}\cdot{{C}_{{{1}}}^{{{8}}}}\right)}\)

\(\displaystyle={\left({1}\cdot{56}\right)}+{\left({5}\cdot{28}\right)}+{\left({10}\cdot{8}\right)}\)

\(\displaystyle={276}\)