Answer to Question #320061 in Statistics and Probability for Feitan

Question

A shipment of 12 Apple mobile phones contains 3 defective phones. In how many ways can a store purchase 5 of these phones and receive at least 2 defective phones?

Answer 1

At least 2 defective phones means one of the two possible outcomes:

1) 2 of 3 defective and 1 of 9 non-defective, or

2) 3 of 3 defective and 0 of 9 non-defective.

As the order of phones does not matter, we'll count the number of combinations without repetition:

$N=\begin{pmatrix} 3 \\ 2 \end{pmatrix} \cdot \begin{pmatrix} 9 \\ 1 \end{pmatrix}+\begin{pmatrix} 3 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 9 \\ 0 \end{pmatrix}=\\ =\cfrac{3!}{2!\cdot(3-2)!}\cdot\cfrac{9!}{1!\cdot(9-1)!}+\\+ \cfrac{3!}{3!\cdot(3-3)!}\cdot\cfrac{9!}{0!\cdot(9-0)!}=\\ =3\cdot9+1\cdot1=28.$