Question

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A , B , C , D , E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $α$ × 56, then $α$ is equal to _______.

Answer 1

If password is 6 character long, tehn

Total number of ways having atleast one number = 106 – 56

Similarly, if 7 character long = 107 – 57

and if 8-character long = 108 – 58

$\begin{array}{l}\text{Number of password = }(10^6 + 10^7 + 10^8) – (5^6 + 5^7 + 5^8) \end{array}$

$\begin{array}{l}= 5^6 \left(2^6 + 5.2^7 + 25.2^8 – 1 – 5 – 25\right)\end{array}$

$\begin{array}{l}= 56\left(64 + 640 + 6400 – 31\right)\end{array}$

$\begin{array}{l}= 7073 \times 5^6\\\\\therefore \alpha = 7073\end{array}$