There is a 20% probability that a person inoculated with a particular vaccine will get the disease anyway. a county health office inoculates 83 people. What is the probabythat exactly 10 of them will get the disease at some point their lives?

The probability is 0.021 or 2.1%.

This is a binomial distribution, since there are two outcomes (infected or not infected), the probabilities are independent of each other, and there is a fixed number of trials.  This is given by:

[tex]_{83}C_{10}\times(0.2)^{10}\times(0.8)^{73} \\ \\=\frac{83!}{10!73!}\times(0.2)^{10}\times(0.8)^{73} \\ \\=0.021[/tex]