A teacher is choosing 4 students from a class of 30 to represent the class at a science fair. In how many ways can the teacher choose the students?
30 c 4
To solve this problem, we need to use the concept of combinations. A combination is a selection of items where the order does not matter.
In this case, we need to choose 4 students from a class of 30. The order of selection does not matter, so we need to find the number of combinations.
The formula to calculate the number of combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of items, r is the number of items to be selected, and ! denotes the factorial of a number.
In this case, n = 30 (total number of students) and r = 4 (number of students to be selected).
Plugging these values into the formula, we get:
C(30, 4) = 30! / (4! * (30 - 4)!)
Simplifying further:
C(30, 4) = 30! / (4! * 26!)
Now, we need to compute the factorials.
30! = 30 * 29 * 28 * ... * 3 * 2 * 1
4! = 4 * 3 * 2 * 1
26! = 26 * 25 * ... * 3 * 2 * 1
We can cancel out some common terms:
C(30, 4) = (30 * 29 * 28 * 27) / (4 * 3 * 2 * 1)
Canceling out further:
C(30, 4) = 27 * 29 * 7
Therefore, the teacher can choose the students in 27 * 29 * 7 = 5,397 ways.