3. A child is playing with letters from a cut-out alphabet. What is the probability that, by arranging the letters К, И, Р, Д, А, Н, З, П in a row, they will form the word ПРАЗДНИК?
S o l u t i o n. Let's denote the event: $A$ - the child will form the word IIPAZDNIC.

Let's find the probability of event $A$, applying formula (1).

The total number of $n$ outcomes of the trial will be obtained using the formulas of combination theory. There are 8 elements - 8 letters; in the formation of different combinations, all 8 elements participate; different combinations differ only in the order of elements; therefore, these combinations are permutations of 8 elements. By formula (2) we get: $n=P_{8}=8!=40320$.

The number of outcomes of the trial favorable to event $A$ is $m=1$, since it is required to form a word with letters arranged in a specific order, and these letters are distinct.

The probability of event $A$ is $P(A)=1 / 40320$.