Answered

9-3: Permutations and Combinations

You download 6 songs on your music player. If you play the songs using the random shuffle option, how many different ways can the sequence of songs be played?

A. 180
B. 360
C. 540
D. 720



Answer :

To determine the number of different ways the sequence of 6 songs can be played, we use the concept of permutations. When you want to find the number of ways to arrange [tex]\( n \)[/tex] distinct items, you use the factorial function, denoted as [tex]\( n! \)[/tex]. The factorial of a number [tex]\( n \)[/tex] is the product of all positive integers from 1 to [tex]\( n \)[/tex].

For example, the factorial of 6 (denoted as [tex]\( 6! \)[/tex]) is calculated as follows:

[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]

Calculating this step-by-step:
- [tex]\( 6 \times 5 = 30 \)[/tex]
- [tex]\( 30 \times 4 = 120 \)[/tex]
- [tex]\( 120 \times 3 = 360 \)[/tex]
- [tex]\( 360 \times 2 = 720 \)[/tex]
- [tex]\( 720 \times 1 = 720 \)[/tex]

Thus, the total number of different ways to arrange or play the 6 songs in sequence is:

[tex]\[ 6! = 720 \][/tex]

Therefore, the correct answer is:
D. 720