Question

How many ways can a teacher select a group of 6 students to sit in the front row if the class has 14 students?

Answer 1

There are 3003 different ways for a teacher to select a group of 6 students to sit in the front row from a class of 14 students, calculated using the combinations formula.

Step-by-step explanation:

The question asks about the number of ways a teacher can select a group of 6 students to sit in the front row from a class of 14 students. This problem can be solved using the concept of combinations, which does not account for the order in which the students are selected. To calculate this, we use the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose.

Applying this formula to our problem, we have n=14 and k=6:

C(14, 6) = 14! / (6!(14-6)!) = 14! / (6!8!) = 3003

Thus, there are 3003 different ways for the teacher to select 6 students out of a class of 14 to sit in the front row.

Answer 2

The question is a combinatorial problem and can be solved using the combination formula. By applying the formula C(14, 6) = 14! / (6! × 8!), we find that there are 3003 ways to select 6 students from 14 to sit in the front row.

Step-by-step explanation:

The question is asking for the number of ways a teacher can select a group of 6 students from a larger group of 14 students to sit in the front row. This type of problem is known as a combinatorial problem in mathematics, and it can be solved using the combination formula. The combination formula is given by C(n, r) = n! / (r! * (n - r)!), where n is the total number of items to choose from, r is the number of items to choose, and ! denotes factorial. In this instance, n is 14 and r is 6.

Step-by-Step Calculation:

Identify the values: n = 14 (total students), r = 6 (students to select).

Calculate the factorial of n: 14! = 14 × 13 × ... × 1.

Calculate the factorial of r: 6! = 6 × 5 × ... × 1.

Calculate the factorial of n - r: 8! = 8 × 7 × ... × 1.

Apply the combination formula: C(14, 6) = 14! / (6! × 8!).

Simplify the factorials and calculate the result to get the number of combinations.

After completing these steps, we can determine that there are 3003 different ways for the teacher to select a group of 6 students out of the class of 14 students to sit in the front row.