To calculate the number of ways to select a committee of 4 members from a club of 12, we use the combination formula. Combinations are used here because the order of selection does not matter.
The formula for combinations is given by:
C(n, k) = n! / (k! * (n – k)!)
Where:
- n is the total number of items (in this case, members in the club).
- k is the number of items to choose (the members on the committee).
- ! represents factorial; for example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Substituting the values into the combination formula, we have:
C(12, 4) = 12! / (4! * (12 – 4)!)
This simplifies to:
C(12, 4) = 12! / (4! * 8!)
Calculating the factorials:
12! = 12 × 11 × 10 × 9 × 8!
Hence, the equation becomes:
C(12, 4) = (12 × 11 × 10 × 9 × 8!) / (4! × 8!)
We can cancel out the 8! in the numerator and denominator:
C(12, 4) = (12 × 11 × 10 × 9) / 4!
Now, calculating 4!:
4! = 4 × 3 × 2 × 1 = 24
Now plug this back into our equation:
C(12, 4) = (12 × 11 × 10 × 9) / 24
Calculating the numerator:
12 × 11 = 132
132 × 10 = 1320
1320 × 9 = 11880
Now dividing by 24:
11880 / 24 = 495
So, the total number of ways to select a committee of 4 from the club of 12 members is 495.