Determining the Number of Groups to Answer Phone Lines in an Office Setting

Imagine a scenario where there are 12 people working in an office with 3 different phone lines. The challenge is to determine how many different groups of 3 people can be formed to answer these phone lines simultaneously. This problem can be effectively solved using the concept of combinations, which is essential in permutation and combination theory.

Understanding the Problem and Applying the Combination Formula

The problem at hand involves finding the number of ways to choose 3 individuals out of 12 people. This is a classic example of a combination problem, where the order of selection does not matter. To solve this, we use the combination formula:

Cnr n! / (r! (n - r)!)

Let's break down the formula and apply it to our scenario:

Defining the Variables

- n (total number of people) 12 - r (number of people to answer the phone lines) 3

Substituting the Values into the Formula

Using the formula, we have:

C123 12! / (3! (12 - 3)!)

Breaking down the factorials:

C123 12! / (3! times; 9!)

Simplifying the expression:

C123 (12 times; 11 times; 10) / (3 times; 2 times; 1)

Calculating the final value:

C123 1320 / 6 220

Conclusion

Therefore, the number of different groups of 3 people that can answer the 3 phone lines is 220. This solution is based on the principle of combinations, where the order in which people answer the lines does not matter. The method used here is a fundamental concept in combinatorial mathematics and is widely used in various scenarios, including scheduling and resource allocation in a workplace environment.

Alternative Approaches to the Problem

Some alternative methods to solve this problem can be:

Permutations Approach

One might argue that there are 12 ways to choose the person for the first phone line, 11 for the second, and 10 for the third. Therefore, the total number of permutations is:

12 times; 11 times; 10 1320

However, this method considers the order, which is unnecessary since the order of answering the lines does not matter. Thus, we need to divide by the number of permutations of the 3 people, which is 3! (3 factorial).

1320 / 3! 1320 / 6 220

Person-by-Person Approach

Another approach is to assign roles based on individual actions. Each phone line can be answered by one individual, and there are 12 people. The first line can have 12 choices, the second line 11 choices, and the third line 10 choices. However, again, this method counts the order, so we need to divide by the number of ways to arrange 3 people, which is 3!.

12 times; 11 times; 10 / 6 220

Conclusion

Regardless of the approach used, the final answer remains the same: 220 possible groups of 3 people to answer the 3 phone lines.