# Suppose a class consists of 5 students majoring in Computer Science, 5 students majoring in Chemistry and 3 students majoring in Mathematics. How many ways are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science?

Question
Suppose a class consists of 5 students majoring in Computer Science, 5 students majoring in Chemistry and 3 students majoring in Mathematics. How many ways are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science?

2021-01-05
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
$$\displaystyle={\left({{C}_{{{0}}}^{{{5}}}}\cdot{{C}_{{{3}}}^{{{8}}}}\right)}+{\left({{C}_{{{1}}}^{{{5}}}}\cdot{{C}_{{{2}}}^{{{8}}}}\right)}+{\left({{C}_{{{2}}}^{{{5}}}}\cdot{{C}_{{{1}}}^{{{8}}}}\right)}$$
$$\displaystyle={\left({1}\cdot{56}\right)}+{\left({5}\cdot{28}\right)}+{\left({10}\cdot{8}\right)}$$
$$\displaystyle={276}$$

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This is how I answered it. P($$\displaystyle{X}_{{1}}$$ and $$\displaystyle{Y}_{{2}}$$) $$\displaystyle={P}{\left({X}_{{1}}\right)}\times{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}={.75}\times{.40}={0.3}.$$
What I don't understand is how do you get the $$\displaystyle{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}$$? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?