JEE MAIN - Mathematics (2022 - 28th July Morning Shift - No. 16)
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 $$\alpha \times 5^{6}$$, then $$\alpha$$ is equal to ___________.
Answer
7073
Explanation
If password is 6 character long, then
Total number of ways having atleast one number $$ = {10^6} - {5^6}$$
Similarly, if 7 character long $$ = {10^7} - {5^7}$$
and if 8-character long $$ = {10^8} - {5^8}$$
Number of password $$ = ({10^6} + {10^7} + {10^8}) - ({5^6} + {5^7} + {5^8})$$
$$ = {5^6}({2^6} + {5.2^7} + {25.2^8} - 1 - 5 - 25)$$
$$ = {5^6}(64 + 640 + 6400 - 31)$$
$$ = 7073 \times {5^6}$$
$$\therefore$$ $$\alpha = 7073$$.
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