Question

There are 16 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

To find out how many groups of 4 people can answer the phone lines in an office of 16, we use the combinations formula. The calculation results in 1820 different groups of 4 people.

Step-by-step explanation:

The student's question is related to combinatorics, a field of mathematics. To find out how many groups of 4 people can answer the phone lines, we consider the number of combinations possible from the 16 people available to answer 4 different phone lines. This type of problem is solved using the combinations formula, which in mathematics is represented as C(n, k) where n is the total number of items, and k is the number of items to choose.

The formula for combinations is C(n, k) = n! / (k! * (n-k)!), where '!' denotes factorial, which is the product of all positive integers up to that number. For this problem, we have n = 16 and k = 4. Thus, the number of ways to choose 4 people out of 16 is C(16, 4) = 16! / (4! * (16-4)!) = 16! / (4! * 12!) = (16 * 15 * 14 * 13) / (4 * 3 * 2 * 1) = 1820.

So, 1820 different groups of 4 people can answer the phone lines.