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Mathematics-Online problems:

Interactive Problem 355: Combinatorics of PINs

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

In a PIN consisting of 5 characters, at least two special symbols ($ \clubsuit$, $ \spadesuit$, $ \heartsuit$) are used apart from the digits 0 to 9 due to security reasons.

        a) How many such PINs do exist?

    How many of these have

        b) only different special characters, e. g. $ \clubsuit$0 $ \heartsuit$9 $ \spadesuit$,
        c) at least two identical special characters, e. g. $ \heartsuit$$ \clubsuit$5 $ \heartsuit$ $ \heartsuit$,
        d) at least two identical digits, e. g. 3$ \clubsuit$3 $ \spadesuit$3 ?


Answer:

        a)  
        b)  
        c)  
        d)  


   

(From: Mathematics Contest 2012)
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  automatically generated: 2/ 6/2018