Management selection. A corporation plans to fill 2 different positions for vice-president, $V_{1}$ and $V_{2}$, from administrative officers in 2 of its manufacturing plants. Plant $A$ has 5 officers and plant $B$ has 9. a. How many ways can these 2 positions be filled if the $V_{1}$ position is to be filled from plant $A$ and the $V_{2}$ position from plant B? b. How many ways can the 2 positions be filled if the selection is made without regard to plant?

Step 1 ：Given that plant A has 5 officers and plant B has 9 officers, we are asked to find the number of ways to fill two different positions for vice-president, $V_{1}$ and $V_{2}$.

Step 2 ：For part a, the $V_{1}$ position is to be filled from plant A and the $V_{2}$ position from plant B. This is a simple permutation problem, as the order of selection matters (i.e., $V_{1}$ and $V_{2}$ are distinct positions). The number of ways to fill the $V_{1}$ position from plant A is the number of officers in plant A, and similarly for the $V_{2}$ position and plant B. Therefore, the total number of ways is the product of the number of officers in each plant, which is $5 \times 9 = 45$.

Step 3 ：For part b, the positions can be filled without regard to plant. This is a combination problem, as the order of selection does not matter. The total number of officers is the sum of the number of officers in plant A and plant B, which is $5 + 9 = 14$. The number of ways to select 2 officers from this total is the combination of the total number of officers taken 2 at a time, which is $\binom{14}{2} = 91$.

Step 4 ：Final Answer: For part a, the number of ways to fill the $V_{1}$ position from plant A and the $V_{2}$ position from plant B is \(\boxed{45}\). For part b, the number of ways to fill the 2 positions without regard to plant is \(\boxed{91}\).