To find out how many ways we can select a committee of 4 members from a club with 12 members, we can use the concept of combinations in combinatorics. When selecting a group where the order of selection does not matter, we use the combination formula:
C(n, r) = n! / (r! * (n – r)!)
In this formula:
- n is the total number of items to choose from (in this case, 12 members).
- r is the number of items to choose (in this case, 4 members).
- ! represents factorial, which means the product of all positive integers up to that number.
Using our values:
- n = 12
- r = 4
We can plug these numbers into our combination formula:
C(12, 4) = 12! / (4! * (12 – 4)!) = 12! / (4! * 8!)
Now, let’s calculate the factorials:
- 12! = 12 × 11 × 10 × 9 × 8! (we can cancel out the 8!)
- 4! = 4 × 3 × 2 × 1 = 24
Substituting back into the formula, we obtain:
C(12, 4) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1)
Calculating the numerator:
- 12 × 11 = 132
- 132 × 10 = 1320
- 1320 × 9 = 11880
Now for the denominator:
4! = 24
Finally, divide the numerator by the denominator:
C(12, 4) = 11880 / 24 = 495
So, there are 495 ways to select a committee of 4 members from a club with 12 members. This means you have plenty of options to form your committee!