4. There are 8 cards; one side of each is clean, while the other side has the letters: И, Я, Л, З, Г, О, О, О printed on them. The cards are placed on the table with the clean side up, shuffled, and then sequentially flipped over one by one. What is the probability that the letters will form the word ЗОоЛОГИЯ when they appear in sequence?
Solution. Let us denote the event: $\boldsymbol{B}$ - the word $300-$ logy will be formed.

We will find the probability of event $B$ by applying formula (1). The numbers $m$ and $n$, included in this formula, will be determined using the formulas of combination theory.

The total number of outcomes of the experiment, as in problem 3, is $n=P_{8}=8$! $=$ $=40320$.

We will number all the cards according to the positions occupied by the letters in the word ZOOLOGY (Fig.1). We will assume that the letters Z, L, G, I, Y are printed on cards $1,4,6,7,8$ respectively. The letter O is printed on cards 2,3 and 5. We will fix the letters Z, L, G, I, Y in positions $1; 4,6,7,8$, and we will change the positions of cards 2,3 and 5 (variants: $2-3-5; 2-5-3; 5-3-2$; $5-2-3; 3-2-5; 3-5-2$). As a result of such changes, we will obtain the word ZOOLOGY. Thus, the number of outcomes of the experiment favorable to event $B$ is $m=P_{3}=3!=6$.

| Letters: | Z | O O L O O I Y |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Numbers of positions: | 12 | 3 | 4 | 5 | 6 | 7 | 8 |

Fig. 1

The probability of event $B$ is $P(B)=6 / 40320=1 / 6720$.