Question

There are 10 students in a class. A teacher would like to split all students into three teams. In how many ways can the teacher do this if each team must have at least three students?

(A) 45 ways
(B) 120 ways
(C) 252 ways
(D) 360 ways

Answer 1

In 252 ways the teacher can do this if each team must have at least three students

Therefore, correct answer is (C) 252 ways

Step-by-step explanation:

To calculate the number of ways the teacher can split the 10 students into three teams, each having at least three students, we can use combinations. The teacher needs to select three students for each team, which can be done using the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of students, and k is the number of students to be selected for each team. In this case, C(10, 3) represents the number of ways to choose three students from 10 for the first team. Since the order of selection doesn't matter, we use combinations. Since there are three teams, we multiply this by C(7, 3) for the second team. Therefore, the total number of ways is C(10, 3) x C(7, 3) x C(4, 3) = 252 ways.

Combinations are an essential concept in combinatorics, helping solve problems related to selecting a certain number of items from a larger set without considering the order. Understanding combinatorics is valuable in various fields, including probability, statistics, and computer science.

Therefore, correct answer is (C) 252 ways