# Suppose a class consists of 4 students majoring in Mathematics, 3 students majoring in Chemistry and 4 students majoring in Computer Science. How many compositions 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 4 students majoring in Mathematics, 3 students majoring in Chemistry and 4 students majoring in Computer Science. How many compositions are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science?

2020-12-25
Total students = 4 + 3 + 4 = 11
Numbers of students majoring in Computer Science = 4
Numbers of students not majoring in Computer Science = 11 - 4 = 7
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 $$\displaystyle={\left({{C}_{{{0}}}^{{{4}}}}\cdot{{C}_{{{3}}}^{{{7}}}}\right)}+{\left({{C}_{{{1}}}^{{{4}}}}\cdot{{C}_{{{2}}}^{{{7}}}}\right)}+{\left({{C}_{{{2}}}^{{{4}}}}\cdot{{C}_{{{1}}}^{{{7}}}}\right)}$$
$$\displaystyle={\left({1}\cdot{35}\right)}+{\left({4}\cdot{21}\right)}+{\left({6}\cdot{7}\right)}$$
$$\displaystyle={161}$$

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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?