Question

The students in group 1 earned 10 extra minutes of recess by winning a class competition. Before going out for their extra recess time, they form a single file line. In how many ways can they line up?

Answer 1

10! = 3628800 ways

Explanation:

Some interesting points

This is a problem of permutations where, at the time of forming different groups, the order of the elements matters but none of them can appear more than once in the same group (no repetition is allowed).

Order matters in the sense that you can form different groups when one element of it changes its position. For instance, suppose having a group of ten students, like the case in the question, whose names are:

`\\ G10 = \{John, Alice, Richard, Ted, Brian, Lisa, Thomas, Virginia, Lincoln, Elizabeth\}`

Then, if we change one student from one position to another---order matters---we have a different group, a permutation, another way of lining them up, like the next one:

`\\ G10 = \{Alice, John, Richard, Ted, Brian, Lisa, Thomas, Virginia, Lincoln, Elizabeth\}` or

`\\ G10 = \{Alice, Elizabeth, Richard, Ted, Brian, Lisa, Thomas, Virginia, Lincoln, John \}`

And so on. Notice that in the file does not appear the same person more than once. In this real case, it is impossible to have the same person twice in the file, so that restriction is perfectly described using permutations.

Solving the question

We can say that we first have ten (10) possible students for occupying the first place. After that, remains nine (9) of them for being in this ninth place, and so on for
`\\ 8^(th), 7^(th), 6^(th), 5^(th), 4^(th), 3^(th), 2^(th), 1^(th)` places.

After knowing this, we need to apply the Multiplication Principle, which roughly says that if we have m possibilities for doing any action and n possibilities for doing another action, then, there are m * n ways of performing both actions.

We can express the former reasons mathematically as follows:

`10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3628800` ways of lining up ten different students in a file.

Notice that

`10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = n!`, which is the factorial of n.

Answer 2

Full Question:

Mrs. Steffen's third grade class has 30 students in it. The students are divided into three groups (numbered 1, 2, and 3), each having 10 students. (a) The students in group 1 earned 10 extra minutes of recess by winning a class competition. Before going out for their extra recess time, they form a single file line. In how many ways can they line up? (b) When all 30 students come in from recess together, they again form a single file line. However, this time the students are arranged so that the first student is from group 1, the second from group 2, the third from group 3, and from there on, the students continue to alternate by group in this order. In how many ways can they line up to come in from recess?

Answer:

1. 3628800 ways

2. 10886400

Step-by-step explanation:

A.

If group 1 students earn 10 extra minute for recess.

A student in group 1 have 1 of 10 positions to fill in

Another student has 9 possible positions

Another has 8 possible positions

Till it gets to the last

So, The 10 students have 10! Number of ways to line up

10! = 3628800

B.

Each of the group have 30 positions to line up

But they must be lined up in 3s (to accommodate each group)

Group 1 has 10! Ways of lining up;

Group 2 has 10! Ways

Group 3 has 10! Ways

So, possible arrangements = 10! * 3

= 10886400