here suppose we have 5 different characters on the keypad

then number of different passwords generated = 5! = 120

chance of guessing the correct one. = 1/120

the question asks us that the probability should be less than 0.01% = 10^-4

so you know with n keypad characters the chance you have is

1/n! and this should be less than 10^-4

solve the inequality: 1/n! < 10^-4, means n! > 10000 and the least n is 8

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Thank you so much!