Three out of seven students in the cafeteria line are chosen to answer survey questions. How many different combinations of three students are possible?
${ }_x c_3=\frac{71}{(7-3) 31}$
What is the answer?
A. 7
B. 35
C. 70
D. 210
Answer (1)
Recognize the problem as a combination since the order of selection doesn't matter.
Apply the combination formula: C ( n , k ) = k ! ( n − k )! n ! , where n = 7 and k = 3 .
Calculate C ( 7 , 3 ) = 3 ! 4 ! 7 ! = 3 × 2 × 1 7 × 6 × 5 .
Simplify to find the number of combinations: 35 .
Explanation
Understand the problem We have 7 students, and we want to choose 3 of them to answer survey questions. Since the order in which we choose the students doesn't matter, this is a combination problem. We need to find the number of combinations of choosing 3 students from a group of 7.
State the combination formula The formula for combinations is given by: C ( n , k ) = k ! ( n − k )! n ! where n is the total number of items, k is the number of items to choose, and ! denotes the factorial. In this case, n = 7 and k = 3 .
Plug in the values Now, we plug in the values of n and k into the formula: C ( 7 , 3 ) = 3 ! ( 7 − 3 )! 7 ! = 3 ! 4 ! 7 ! This means we need to calculate the factorials of 7, 3, and 4.
Calculate the factorials Let's expand the factorials: 7 ! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 3 ! = 3 × 2 × 1 = 6 4 ! = 4 × 3 × 2 × 1 = 24 So, we have: C ( 7 , 3 ) = 6 × 24 5040 = 144 5040
Simplify and calculate Now, we simplify the expression: C ( 7 , 3 ) = ( 3 × 2 × 1 ) ( 4 × 3 × 2 × 1 ) 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3 × 2 × 1 7 × 6 × 5 = 6 7 × 6 × 5 C ( 7 , 3 ) = 7 × 5 = 35 Therefore, there are 35 different combinations of three students that can be chosen from the seven students.
Final Answer Thus, the number of different combinations of three students that can be chosen from the seven students is 35.
Examples
In a classroom of 7 students, a teacher wants to form a group of 3 students for a project. The number of different groups that can be formed is a combination problem. This concept is useful in scenarios like forming committees, selecting teams, or choosing a subset of items from a larger set where the order of selection doesn't matter. For example, if you have 7 different ingredients and you want to make a salad with only 3 ingredients, you can use combinations to find out how many different salads you can make.
