To determine how many different combinations of 4 students can be selected from a group of 12 students, we use the concept of combinations in combinatorics. 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:
- C(n, r) is the number of combinations,
- n is the total number of items (in this case, students),
- r is the number of items to choose, and
- ! denotes factorial, the product of all positive integers up to that number.
For this specific problem:
- n = 12 (the total number of students)
- r = 4 (the number of students to select)
Now, we can substitute the values into the formula:
C(12, 4) = 12! / (4! * (12 – 4)!)
This simplifies to:
C(12, 4) = 12! / (4! * 8!)
Next, we can expand the factorials:
12! = 12 × 11 × 10 × 9 × 8!
Notice that the 8! in the numerator and denominator will cancel each other out, giving us:
C(12, 4) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1)
Calculating the numerator:
- 12 × 11 = 132
- 132 × 10 = 1320
- 1320 × 9 = 11880
This gives us a numerator of 11880.
Now for the denominator:
- 4 × 3 = 12
- 12 × 2 = 24
- 24 × 1 = 24
Now we divide the numerator by the denominator:
C(12, 4) = 11880 / 24 = 495
Therefore, there are 495 different combinations of 4 students that can be selected from a group of 12 students to serve on the university committee.