In how many ways can a teacher arrange 6 students in the front row of a classroom with a total of 20 students
what is
20x19x18x17x16x15 ?
27,907,200
To find the number of ways the teacher can arrange 6 students in the front row, we can use the concept of permutations.
The number of ways to arrange 6 students out of a total of 20 can be calculated using the permutation formula:
P(n, r) = n! / (n - r)!
Where n is the total number of objects (students) and r is the number of objects (students) to be arranged (6 in this case).
Plugging in the values, we have:
P(20, 6) = 20! / (20 - 6)!
= 20! / 14!
= (20 * 19 * 18 * 17 * 16 * 15 * 14!) / 14!
= 20 * 19 * 18 * 17 * 16 * 15
= 9,362,880
Therefore, the teacher can arrange the 6 students in the front row in 9,362,880 different ways.
To determine the number of ways the teacher can arrange 6 students in the front row, we can use the concept of permutations. In permutations, the order of the arrangement matters. In this case, we want to choose 6 students from a total of 20.
The formula to calculate permutations is:
P(n, r) = n! / (n - r)!
Where P is the permutation function, n is the total number of students, and r is the number of students in the front row.
Using this formula, we can substitute in the values for our question:
P(20, 6) = 20! / (20 - 6)!
Calculating this using factorial notation, we get:
P(20, 6) = 20! / 14!
Now, calculating the factorials:
P(20, 6) = (20 × 19 × 18 × 17 × 16 × 15 × 14!) / 14!
The 14! terms cancel out:
P(20, 6) = 20 × 19 × 18 × 17 × 16 × 15
Evaluating this expression, we find:
P(20, 6) = 27,907,200
Therefore, there are 27,907,200 ways the teacher can arrange 6 students in the front row.