A license plate has a sequence of digits of length n. Each digit 0 through 9 may occur in any place in the sequence.
1) Given n = 7, what fraction of all license plates with no repeated digits start with its largest digit?
2)Given n = 7, consider all license plates with no repeated digits that start with its largest digit. What fraction of these end with its second largest digit?
How do you factor in probability for the plate starting with the largest number when it does not have to be the largest number available, just the largest number in the sequence?
$1)$ We're given that the license place has $7$ distinct digits, so we just need to find the probability that of those $7$ digits, the largest one is in the starting position. This probability is $\frac{1}{7}$
$2)$ We're given that the license place has $7$ distinct digits and that the first digit is the largest. Now we have $6$ positions and want to find the probability that it ends with the new largest digit. This probability is $\frac{1}{6}$.