To determine the number of ways a television programming director can schedule six different commercials in the six time slots allocated during a 1-hour program, we use the concept of permutations.
A permutation is an arrangement of all the members of a set into a specific sequence or order.
In this case, we have 6 different commercials (let's denote this as [tex]\( n = 6 \)[/tex]) and we need to arrange all 6 commercials in 6 time slots (so [tex]\( r = 6 \)[/tex]).
To find the number of permutations (denoted as [tex]\( nP_r \)[/tex]), we use the formula:
[tex]\[ {}_nP_r = \frac{n!}{(n-r)!} \][/tex]
Plugging in our values [tex]\( n = 6 \)[/tex] and [tex]\( r = 6 \)[/tex]:
[tex]\[ {}_6P_6 = \frac{6!}{(6-6)!} \][/tex]
Simplify the factorial terms:
[tex]\[ (6-6)! = 0! \][/tex]
Given from the hint, we know that:
[tex]\[ 0! = 1 \][/tex]
Thus:
[tex]\[ {}_6P_6 = \frac{6!}{1} \][/tex]
Now, we need to recall the value of [tex]\( 6! \)[/tex]. The factorial of 6 is calculated as:
[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \][/tex]
Therefore:
[tex]\[ \frac{6!}{1} = 720 \][/tex]
So, the number of ways the television programming director can schedule the six different commercials in six time slots is:
[tex]\[ 720 \][/tex]
Hence, the total number of possible schedules is [tex]\( 720 \)[/tex].
