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Mathematics-Online problems:

Solution to the problem of the (previous) week

Problem:

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)  

Solution:

a)     Determining the total number of PINs, we group them according to their number of special characters (SC) and obtain:

                                2 SC: $ \left( \begin{array}{c} 5 \\ 2 \end{array} \right) \cdot 3^2 \cdot 1000$ = $ 90 000$
                                3 SC: $ \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \cdot 3^3 \cdot 100$ = $ 27 000$
                                4 SC: $ \left( \begin{array}{c} 5 \\ 4 \end{array} \right) \cdot 3^4 \cdot 10$ = $ 4 050$
                                5 SC: $ \left( \begin{array}{c} 5 \\ 5 \end{array} \right) \cdot 3^5 \cdot 1$ = $ 243$
                                So, the total number is $ 121 293$


b)      If only different special characters are allowed, we get a modified table:

                                2 SC: $ \left( \begin{array}{c} 5 \\ 2 \end{array} \right) \cdot 3 \cdot 2 \cdot 1000 $ = $ 60 000 $
                                3 SC: $ \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \cdot 3 \cdot 2 \cdot 1 \cdot 100 $ = $ 6 000 $
                                i. e., there exist $ 66 000$ of these PINs.


c)      From the total number of PINs, we have to subtract those with only different special characters. Hence, there are
         $ 121 293 - 66 000 = 55 293$ of such PINs.
         This result can also be obtained, if in each case of table a) we subtract the number of PINs with only different special characters:

                                2 SC: $ \left( \begin{array}{c} 5 \\ 2 \end{array} \right) \cdot \left[3^2 - 3 \cdot 2\right] \cdot 1000$ = $ 30 000$
                                3 SC: $ \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \cdot \left[3^3 - 3 \cdot 2 \cdot 1\right] \cdot 100$ = $ 21 000$
                                4 SC: $ \left( \begin{array}{c} 5 \\ 4 \end{array} \right) \cdot \left[3^4 - 0 \right] \cdot 10$ = $ 4 050$
                                5 SC: $ \left( \begin{array}{c} 5 \\ 5 \end{array} \right) \cdot \left[3^5 - 0 \right] \cdot 1$ = $ 243$
                                So, these numbers amount to $ 55 293$


d)      Analogous to step c) we obtain

                                2 SC: $ \left( \begin{array}{c} 5 \\ 2 \end{array} \right) \cdot 3^2 \cdot (1000 - 10 \cdot 9 \cdot 8)$ = $ 25 200$
                                3 SC: $ \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \cdot 3^3 \cdot (100 - 10 \cdot 9)$ = $ 2 700$
                                the sum being $ 27 900$

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