Content
A number has a peak if the largest digit is not at either end of the number, such that all the digits to its left are in increasing order and all the digits to its right are in decreasing order. For example, the numbers
have a peak. On the other hand, 154623 and 123456 (the 6 is at the end) do not contain a peak.
A peak password is a number made up of the digits from 1 through 9, where each digit is used exactly once, and contains a peak.
Explain why the peak of such a password must be 9.
Now consider all the passwords whose peak is the 4th digit from the left.
Show that the number of such passwords can be written in the form (n3), and find the value of n.
Using a specifically chosen value of x and the expansion of (1+x)n, show that
Using the result from (c), determine the total number of possible peak passwords.
Ask Plex AI about this problem
Get hints, ask questions, and work through this problem step by step