How do I solve this problem for Math 108 Quantative Reasonsing

Question

A student downloaded 4 music files to a portable music player. In how many different orders can the songs be played?

Answer 1

This is called a permutation of 4 items taken 4 at a time.

Picture 4 slots that are going to be filled with one of the songs.

For the 1st slot there are 4 choices. Once the 1st slot is filled, there remain 3 choices for the 2nd slot. This leads to the fact that there will be 2 choices for the 2rd slot. Finally there is only one song left for the last slot.

The fundamental counting principle indicates that the total number of possible outcomes is:

4(3)(2)(1) = 24 different ways to play the 4 songs.

This can be written as 4! (4 factorial) which means 4 x 3 x 2 x 1.

What is you were only going to play 2 of the songs?

Then 4 choices times 3 choices results if 12 different ways to play the song.

₄P₂ : permutations of 4 items taken 2 at a time.

_nP_r : permutations of n items take r at a time has the following formula:

n!/(n - r)!

That denominator is simply a way to stop filling in the slots.

₇P₃ = (7!)/(7 - 3)! = (7x6x5x4x3x2x1)/(4x3x2x1) = 7x6x5, i.e. only filling in 3 slots.

Nest on your plate is likely the difference between permutations and combinations.