The management of a company, a president and four vice-presidents, denoted by the set $\left\{P, V_{1}, V_{2}, V_{3}, V_{4}\right\}$, wish to select a committee of 4 people from among themselves. How many ways can this committee be formed? That is, how many 4-person subsets can be formed from a set of 5 people? How many ways can a 4-person committee be formed using the president and four vice-presidents?

Step 1 ：The problem is asking us to find the number of ways a committee of 4 people can be formed from a group of 5 people, which includes a president and four vice-presidents.

Step 2 ：This is a combination problem, where we are choosing 4 people out of 5. The order does not matter and we are not replacing.

Step 3 ：The formula for combinations is \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of options, k is the number of options chosen, and ! denotes factorial.

Step 4 ：In this case, n=5 (the president and four vice-presidents) and k=4 (the size of the committee).

Step 5 ：Substituting the values into the formula, we get \(C(5, 4) = \frac{5!}{4!(5-4)!}\).

Step 6 ：Solving the above expression, we find that there are 5 ways to form a 4-person committee from a group of 5 people.

Step 7 ：Final Answer: \(\boxed{5}\)