How many different ways can 3 students be selected from a group of 10?

How many different ways can 3 students be selected from a group of 10?

Correct Answer: 120

Step 1: Understand that we want to choose 3 students from a total of 10 students.
Step 2: Recognize that the order in which we select the students does not matter. This means we will use combinations, not permutations.
Step 3: The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
Step 4: In our case, n = 10 (total students) and r = 3 (students to select).
Step 5: Plug the values into the formula: C(10, 3) = 10! / (3! * (10 - 3)!).
Step 6: Simplify the formula: C(10, 3) = 10! / (3! * 7!).
Step 7: Calculate 10! = 10 × 9 × 8 × 7! (we can cancel 7! in the numerator and denominator).
Step 8: Now we have C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1).
Step 9: Calculate the numerator: 10 × 9 × 8 = 720.
Step 10: Calculate the denominator: 3 × 2 × 1 = 6.
Step 11: Divide the numerator by the denominator: 720 / 6 = 120.
Step 12: Therefore, there are 120 different ways to select 3 students from a group of 10.

Combinations – The concept of combinations is used to determine the number of ways to select a subset of items from a larger set, where the order of selection does not matter.