A dance instructor chose four of his 10 students to be on stage for a performance. If order does not matter, in how many different ways can the instructor choose the four students?
Answer (1)
Recognize the problem as a combination.
Apply the combination formula: C ( n , k ) = k ! ( n − k )! n ! .
Substitute n = 10 and k = 4 into the formula.
Calculate the result: 210 .
Explanation
Analyze the problem The problem asks for the number of ways to choose 4 students out of 10, where the order of selection does not matter. This is a combination problem.
Apply the combination formula We use the combination formula: C ( n , k ) = k ! ( n − k )! n ! , where n is the total number of items and k is the number of items to choose. In this case, n = 10 and k = 4 .
Substitute values Substitute the values into the formula: C ( 10 , 4 ) = 4 ! ( 10 − 4 )! 10 ! = 4 ! 6 ! 10 !
Calculate factorials Calculate the factorials: 10 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3 , 628 , 800 , 4 ! = 4 × 3 × 2 × 1 = 24 , and 6 ! = 6 × 5 × 4 × 3 × 2 × 1 = 720 .
Simplify Simplify the expression: C ( 10 , 4 ) = 24 × 720 3 , 628 , 800 = 17 , 280 3 , 628 , 800 = 210 .
Conclusion Therefore, there are 210 different ways to choose the four students.
Examples
In a class of 10 students, a teacher wants to form a committee of 4 students to represent the class in a school event. Since the order in which the students are chosen does not matter, this is a combination problem. The number of ways to form the committee is calculated using the combination formula, which helps the teacher determine the different possible groups of students that can be selected. This ensures fairness and variety in student representation.
