Final answer:
The question involves combinatorial mathematics and asks for the number of ways to lay off employees from different offices. Part A requires a complex combinatorial calculation, while Part B is a calculation of product combinations from each office, using the combination formula.
Step-by-step explanation:
The question relates to combinatorics, which is a branch of mathematics focused on counting combinations and permutations of sets.
Part A: Counting the Number of Ways to Lay Off 10 Employees
To calculate the total number of ways to lay off 10 employees from the three offices with no restrictions, we must consider the combinations from each office. Since there are 5 employees in New York, 7 in London, and 9 in Shanghai, we can model this as a problem of distributing 10 identical layoffs among the three offices without any restrictions other than the number of employees available in each office.
This is a combinatorial problem involving partitions and can be solved using generating functions or other combinatorics techniques. However, the actual solution requires additional information or a clear methodological path, which is beyond the scope of this platform.
Part B: Specified Layoffs in Each Office
For the specific scenario where 2 employees are laid off in New York, 5 in London, and 3 in Shanghai, we can calculate the number of ways by multiplying the combinations from each office. Since combinations don't take the order of selection into account, the number can be calculated as the product of each office's combination. We use the formula for combinations: C(n, k) = n! / (k!(n-k)!), where n is the total number of employees and k is the number of employees to be laid off.
In New York, there are C(5, 2) ways to lay off 2 employees. In London, there are C(7, 5) ways to lay off 5 employees. Lastly, in Shanghai, there are C(9, 3) ways to lay off 3 employees. Therefore, the total number of ways to lay off employees as specified can be calculated as the product of these combinations.