To determine how many ways we can select a committee of four students from a club consisting of 15 members, we need to use the concept of combinations. Combinations are used when the order of selection does not matter.
The formula for combinations is given by:
C(n, r) = n! / (r!(n – r)!)
Where:
- n is the total number of items to choose from (in this case, 15 members).
- r is the number of items to choose (in this case, 4 students).
- ! denotes factorial, which is the product of all positive integers up to a given number.
Using the values:
- n = 15
- r = 4
We can plug these values into the formula:
C(15, 4) = 15! / (4!(15 – 4)!) = 15! / (4! * 11!)
Now we simplify:
- 15! = 15 × 14 × 13 × 12 × 11!
- Thus, C(15, 4) = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1)
Calculating this gives:
- 15 × 14 = 210
- 210 × 13 = 2730
- 2730 × 12 = 32760
- 4 × 3 × 2 × 1 = 24
Now, we divide the two:
C(15, 4) = 32760 / 24 = 1365
Therefore, the number of ways to select a committee of four students from a 15 member club is 1365.