Problem 42 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) A corporation has ten members on its board of directors. In how many different ways can it elect a president, vice president, secretary, and treasurer?

Step-by-Step Solution

Answer

There are 5040 different ways the corporation can elect a president, vice president, secretary, and treasurer from its board of directors.

Step 1: Identifying the values for n and r

In this case, the total number of directors on the board (n) is 10. The number of positions to fill (r) is 4 (president, vice president, secretary, treasurer).

Step 2: Using the permutation formula

Now we'll plug the values into our permutation formula \(_nP_{r} = n!/(n-r)!\). So we have to calculate \( _{10} P_{4} = 10!/(10-4)!\)

Step 3: Calculate factorial and simplify

Next, calculate 10! (10*9*8*7*6*5*4*3*2*1) and 6! (6*5*4*3*2*1), then divide 10! by 6!. The factorial of any number n, denoted by n!, is the product of all positive integers less than or equal to n. It symbolizes the number of ways n objects can be arranged.

Step 4: Final calculation

After calculating, the division gives 5040. This represents the total number of ways the corporation can elect a president, vice president, secretary, and treasurer from its board of directors.