Respuesta :
Answer:
The smallest value of r such that there are more than 2,000,000 possible passwords is r=9.
Step-by-step explanation:
Given : Numeric passwords of length r consist of n digits from {0,1,2,…,9}. Digits may be not repeated (e.g., 1178 is a not a permissible password of length 4).
To Find : The smallest value of r such that the number of possible passwords of length r is greater than 2,000,000.
Solution :
Numeric passwords of length r consist of n digits from {0,1,2,…,9}
i.e. There are 10 possible digits : 0,1,2,3,4,5,6,7,8,9.
So, The first digit have 10 ways,
Second digit have 9 ways different from previous digit.
Third digit have 8 ways different from previous digit.
Similarly, r th digit have n-r+1 ways.
Applying fundamental counting principle,
If the first event occur in m ways and second event occur in n ways the the number of ways two events occur in sequence is [tex]m\cdot n[/tex]
[tex]10\cdot 9\cdot 8\cdot ....\cdot (n-r+1)[/tex] is the required ways.
But The smallest value of r such that the number of possible passwords of length r is greater than 2,000,000.
i.e. [tex]10\cdot 9\cdot 8\cdot ....\cdot (n-r+1)\geq 2000000[/tex]
The increasing value of r will obtain more than 2,000,000 possible passwords were,
If r=1
Number of passwords = 10
If r=2
Number of passwords = [tex]10\cdot 9=90[/tex]
If r=3
Number of passwords = [tex]10\cdot 9\cdot8=720[/tex]
If r=4
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7=5040[/tex]
If r=5
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7\cdot 6=30240[/tex]
If r=6
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7\cdot 6\cdot 5=151200[/tex]
If r=7
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7\cdot 6\cdot 5\cdot 4=604800[/tex]
If r=8
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3=1814400[/tex]
If r=9
Number of passwords = [tex]10\cdot 9\cdot8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2=3628800[/tex]
For r = 9 the password length exceeds.
Therefore, The smallest value of r such that there are more than 2,000,000 possible passwords is r=9.
