Question

Five students in a class have volunteered for a special project. Only three can actually help with the project. How many groupings of three out of the five are possible?

Answer 1

To find the number of possible groupings of three students out of five, a combinatorial calculation using the combination formula is used, resulting in 10 possible groupings.

Step-by-step explanation:

The number of groupings of three out of five students can be calculated using the combination formula. This is a combinatorial problem specifically dealing with combinations since the order in which the students are grouped does not matter.

The combination formula is defined as №(n, k) = n! / (k!(n-k)!), where №(n, k) is the number of combinations, n is the total number of items, and k is the number of items to choose. In this case, n is 5 and k is 3.

To calculate this, follow these steps:

1. Calculate the factorial of n, which is 5! (5 factorial).
2. Calculate the factorial of k, which is 3! (3 factorial).
3. Calculate the factorial of n-k, which is (5-3)! or 2! (2 factorial).
4. Plug these values into the combination formula to get №(5, 3) = 5! / (3!2!).
5. Calculate the result: 5! / (3!2!) = (5×4×3×2×1) / ((3Ô2Ô1)(2Ô1)) = 120 / (6×2) = 120 / 12 = 10.

Therefore, there are 10 possible groupings of three students out of five.