judokan
A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?
(A) 3,024
(B) 4,536
(C) 5,040
(D) 9,000
(E) 10,000
Immediate application of the Multiplicative Principle:
\(\begin{array}{*{20}{c}}\\
{\underline {{\text{not}}\,\,0} } \\ \\
9 \\
\end{array}\begin{array}{*{20}{c}}\\
{\underline {{\text{nr}}} } \\ \\
9 \\
\end{array}\begin{array}{*{20}{c}}\\
{\underline {{\text{nr}}} } \\ \\
8 \\
\end{array}\begin{array}{*{20}{c}}\\
{\underline {{\text{nr}}} } \\ \\
7 \\
\end{array}\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{Multipl}}{\text{.}}\,{\text{Principle}}} \,\,\,\,? = {9^2} \cdot 8 \cdot 7\,\,\,\,\,\,\,\,\,\,\left[ {nr = {\text{no}}\,\,{\text{repetition}}} \right]\)
\(\left\langle ? \right\rangle = \left\langle {{9^2}} \right\rangle \cdot \left\langle {8 \cdot 7} \right\rangle = 1 \cdot 6 = 6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\left\langle N \right\rangle = {\text{units}}\,\,{\text{digit}}\,\,{\text{of}}\,\,N} \right]\)
Just one alternative choice with unit´s digit equal to the correct one... we are done!
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.