197. The device consists of $n$ elements. The probability of failure of any element during the experiment is $p$. Find the mathematical expectation of the number of such experiments, in each of which exactly $m$ elements will fail, if a total of $\boldsymbol{N}$ experiments are conducted. It is assumed that the experiments are independent of each other.
Solution. Let $X$ be the number of trials in which exactly $m$ elements will fail. Since the trials are independent and the probability of the event of interest (exactly $\boldsymbol{m}$ elements will fail in one trial) is the same in these trials, the formula

$$
M(X)=N P
$$

can be applied, where $N$ is the total number of trials; $P$ is the probability that exactly $m$ elements will fail in one trial.

We will find the probability $P$ using the Bernoulli formula:

$$
P=C_{n}^{m} p^{m} q^{n-m}
$$

Substituting (**) into (*), we obtain the desired expected value:

$$
M(X)=N C_{n}^{m} p^{m} q^{n-m}
$$