Question

A student organization lists as its members 2 freshmen, 3 sophomores, 5 juniors, and 2 seniors. If a team of 4 is to be selected at random, find the probability that all year levels are represented. 

Answer 1

There are 12 people in total. The number of combinations we can choose 4 members out of 12 (without any constraints) is

$C(12, 4) = \begin{pmatrix} 12 \\ 4 \end{pmatrix} = 495$

Then among these 4 members, we need to have 1 freshman, 1 sophomore, 1 junior, and 1 senior.. This can be done in

$\begin{pmatrix} 2 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 5 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \end{pmatrix} = 2 \cdot 3 \cdot 5 \cdot 2 =60$

The probability is

$\displaystyle P(A) = \frac{60}{495} = 0.12 =12\%$

Answer: 0.12