Question

There are 9 people in an office with 4 different phone lines. If all the lines begin to ring at once, how many groups of 4 people can answer these lines?

Answer 1

Using the combination formula C(9, 4), which is derived as 9!/(4!(9-4)!), we find that there are 126 different ways for 4 out of 9 people to answer four phone lines.

Step-by-step explanation:

The question requires us to determine how many groups of 4 people can answer four different phone lines if there are 9 people in an office. This is a combinatorial problem that can be solved by using the combinations formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. In this case, n equals 9 (the number of people) and k equals 4 (the phone lines to be answered).

To find the number of combinations of 9 people taken 4 at a time, we calculate C(9, 4):

• First, calculate the factorial for each relevant number: 9! = 9×8×7×6×5×4×3×2×1 = 362,880 and 4! = 4×3×2×1 = 24.
• Subtract the number of items to choose from the total number of items, then find the factorial: (9-4)! = 5! = 5×4×3×2×1 = 120.
• Divide 9! by the product of 4! and 5!: 362,880 / (24 × 120) = 362,880 / 2,880 = 126.

Therefore, there are 126 different ways in which 4 out of 9 people can answer 4 different phone lines.