You download eight songs onto a music player. If you play three of the songs using the random shuffle option, how many ways can the sequence of songs be played?

Note: [tex]${}_nP_r = \frac{n!}{(n-r)!}$[/tex]



Answer :

To determine the number of ways to play three songs out of a total of eight in sequence, we need to calculate the number of permutations of 8 items taken 3 at a time.

The formula for permutations [tex]\( P(n, r) \)[/tex] is given by:

[tex]\[ P(n, r) = \frac{n!}{(n-r)!} \][/tex]

Here, [tex]\( n = 8 \)[/tex] (the total number of songs) and [tex]\( r = 3 \)[/tex] (the number of songs to be played in sequence).

Let's insert the values into the formula:

[tex]\[ P(8, 3) = \frac{8!}{(8-3)!} \][/tex]

This simplifies to:

[tex]\[ P(8, 3) = \frac{8!}{5!} \][/tex]

Since [tex]\( 8! \)[/tex] (8 factorial) is the product of all positive integers up to 8, and [tex]\( 5! \)[/tex] (5 factorial) is the product of all positive integers up to 5, we can write it as:

[tex]\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]

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

Now, [tex]\( 5! \)[/tex] (the product of the numbers from 5 down to 1) will cancel out from both the numerator and the denominator, leaving us with:

[tex]\[ P(8, 3) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} = 8 \times 7 \times 6 \][/tex]

So we calculate:

[tex]\[ 8 \times 7 = 56 \][/tex]

[tex]\[ 56 \times 6 = 336 \][/tex]

Therefore, the number of ways to play three of the eight songs in sequence is:

[tex]\[ \boxed{336} \][/tex]