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
