9. In a batch of $N$ items, there are $k$ standard items. For inspection, $l$ items are randomly selected. Find the probability that among the selected items, exactly $r$ are standard.

翻译结果如上，保留了原文的换行和格式。
S o l u t i o n. Let us denote the event: $A$ - among the selected $l$ items, there are $r$ standard items.

We will find the probability of event $A$ by applying formula (1). The numbers $m$ and $n$, which are part of this formula, will be obtained using the formulas of combination theory.

There are a total of $N$ elements - $N$ items in the batch. These elements are represented in Fig. 6. Standard items are denoted by the symbols $\otimes$, and non-standard items by the symbols $O$. The number of standard items in the batch is $k$, and the number of non-standard items is $N-k$ (Fig. 6, a).

a) Batch of $N$ items:

![](https://cdn.mathpix.com/cropped/2024_05_22_c311c9dd990dc55c716eg-015.jpg?height=121&width=993&top_left_y=827&top_left_x=248)

b) The selected $l$ items for inspection:

$r$ standard items

![](https://cdn.mathpix.com/cropped/2024_05_22_c311c9dd990dc55c716eg-015.jpg?height=71&width=291&top_left_y=1109&top_left_x=318)

$l-r$ non-standard items

![](https://cdn.mathpix.com/cropped/2024_05_22_c311c9dd990dc55c716eg-015.jpg?height=77&width=288&top_left_y=1108&top_left_x=871)

Fig. 6

The total number $n$ of possible outcomes of the trial is equal to the number of ways to select $l$ elements from $N$. Each combination of $N$ elements includes $l$ elements. Different combinations differ from each other by at least one element, and the order of the elements does not matter. Therefore, these combinations represent combinations of $N$ elements taken $l$ at a time. Thus, $n=C_{N}^{l}$.

Let us find the number of outcomes of the trial that are favorable to event $A$. Among the selected $l$ items, there are $r$ standard and $l-r$ non-standard items (Fig. 6, b). We will classify the standard items as the first group of items, and the non-standard items as the second group.

The number of ways to select the required $r$ standard items from the $k$ standard items available in the batch is $s=C_{k}^{r}$. The number of ways to select $l-r$ non-standard items from the $N-k$ such items available in the batch is $t=C_{N-k}^{l-r}$.

Any combination of items from the first group can be combined with any combination of items from the second group. Therefore, the number of outcomes favorable to event $A$ is $m=s \cdot t=C_{k}^{r} \cdot C_{N-k}^{l-r}$.

The probability of event $A$ is

$$
P(A)=\frac{C_{k}^{r} \cdot C_{N-k}^{l-r}}{C_{N}^{l}}
$$