How many ways can 6 people be selected from a group of 10? (2020)

How many ways can 6 people be selected from a group of 10? (2020)

Step 1: Understand that we want to choose 6 people from a group of 10 people.
Step 2: Recognize that this is a combination problem because the order of selection does not matter.
Step 3: Use the combination formula C(n, r) = n! / (r! * (n - r)!), where n is the total number of items (10) and r is the number of items to choose (6).
Step 4: Plug in the values into the formula: C(10, 6) = 10! / (6! * (10 - 6)!)
Step 5: Simplify the formula: C(10, 6) = 10! / (6! * 4!)
Step 6: Calculate 10! = 10 × 9 × 8 × 7 × 6! (we can cancel 6! in the numerator and denominator)
Step 7: Now we have: C(10, 6) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1)
Step 8: Calculate the numerator: 10 × 9 = 90, then 90 × 8 = 720, then 720 × 7 = 5040.
Step 9: Calculate the denominator: 4 × 3 = 12, then 12 × 2 = 24, then 24 × 1 = 24.
Step 10: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 11: Conclude that there are 210 ways to select 6 people from a group of 10.

Combinatorics – The study of counting, arrangements, and combinations of objects.
Binomial Coefficient – The formula C(n, k) = n! / (k!(n-k)!) used to calculate the number of ways to choose k elements from a set of n elements.