To evaluate the expression 9C3, we need to calculate the number of combinations of 9 items taken 3 at a time. This is a common problem in combinatorics, and it can be solved using the combination formula:
Combination Formula:
C(n, k) = n! / (k! * (n – k)!)
Where:
- n is the total number of items.
- k is the number of items to choose.
- ! denotes factorial, which is the product of all positive integers up to that number.
Let’s apply this formula to 9C3:
Step 1: Calculate the factorial of 9 (9!).
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362880
Step 2: Calculate the factorial of 3 (3!).
3! = 3 × 2 × 1 = 6
Step 3: Calculate the factorial of (9 – 3), which is 6!.
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
Step 4: Plug these values into the combination formula.
C(9, 3) = 9! / (3! * (9 – 3)!) = 362880 / (6 * 720) = 362880 / 4320 = 84
Final Answer:
9C3 = 84
So, there are 84 different ways to choose 3 items from a set of 9 items.