A carton contains 17 good apples and 3 rotten apples. A fruit seller picks three apples in random from the carton and puts them into a smaller box for sale.

1. Find the probability that all three apples picked are good

There are $17$ good apples out of $20$ apples in total.

Without replacement,
$P(\text{all 3 good})=\frac{17}{20}\times\frac{16}{19}\times\frac{15}{18}.$

2. Subtract from 1 to get the probability of at least one rotten apple

$P(\text{at least 1 rotten})=1-P(\text{all 3 good})$
$=1-\frac{17}{20}\times\frac{16}{19}\times\frac{15}{18}$
$=1-\frac{34}{57}$
$=\frac{23}{57}.$

1. Identify what Student A's product represents

Student A's product

$\frac{17}{20}\times \frac{16}{19}\times \frac{3}{18}$

represents picking a good apple first, another good apple second, and a rotten apple third, i.e. the order $GGR$ only.

2. Account for all possible orders of two good apples and one rotten apple

The box can contain two good apples and one rotten apple in $3$ possible orders:

$GGR,\ GRG,\ RGG.$

Each order has the same probability, so

$P(\text{2 good and 1 rotten})=3\times \frac{17}{20}\times \frac{16}{19}\times \frac{3}{18}=\frac{34}{95}.$