Question

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?

Answer 1

The number of ways to choose 4 students from a class of 30 to represent the class at a science fair is 27,405, calculated using the combination formula.

Step-by-step explanation:

The question involves combinatorics, which is a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. Specifically, the problem asks, "In how many ways can a teacher choose 4 students from a class of 30 to represent the class at a science fair?" The answer can be found using binomial coefficients, which are the coefficients of the terms in the expansion of a binomial power, in this case, the combination of 30 students taken 4 at a time.

To solve this, we use the combination formula: C(n, k) = n! / (k! * (n-k)!), where n is the total number of items, k is the number to be chosen, n! denotes the factorial of n, and k! denotes the factorial of k.

Therefore, the number of ways to choose 4 students from 30 is:
C(30, 4) = 30! / (4! * (30-4)!) = (30*29*28*27) / (4*3*2*1) = 27,405.

Answer 2

`{30 \choose 4}=(30!)/(4!26!)=(27\cdot28\cdot29\cdot30)/(2\cdot 3\cdot4)=27\cdot7\cdot29\cdot5=27405`