How many ways can 4 students be selected from a group of 10?

How many ways can 4 students be selected from a group of 10?

Correct Answer: 210

Step 1: Understand that we want to choose 4 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 = 4 (students to choose).
Step 5: Plug the values into the formula: C(10, 4) = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!).
Step 6: Calculate 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, but we can simplify it by canceling out 6! in the denominator.
Step 7: This simplifies to C(10, 4) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 × 7 = 5040.
Step 9: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 10: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 11: Therefore, the number of ways to select 4 students from 10 is 210.

Combinatorics – The question tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection.