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In certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64 . How many telephones numbers have all six digits distinct?
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Verified Answer
If first two digits are 41 , the remaining 4 digits can be arranged in
$$
\begin{aligned}
={ }^8 P_4=\frac{8 !}{8-4 !} &=\frac{8 !}{4 !}=\frac{8 \times 7 \times 6 \times 5 \times 4 !}{4 !} \\
&=8 \times 7 \times 6 \times 5=1680
\end{aligned}
$$
Similarly, iffirst two digits are $42,46,62$, or 64 , the remaining 4 digits can be arranged in same way i.e., 1680 ways.
$\therefore$ Total number of telephone numbers having all six digits
$$
\text { distinct }=5 \times 1680=8400 \text {. }
$$
$$
\begin{aligned}
={ }^8 P_4=\frac{8 !}{8-4 !} &=\frac{8 !}{4 !}=\frac{8 \times 7 \times 6 \times 5 \times 4 !}{4 !} \\
&=8 \times 7 \times 6 \times 5=1680
\end{aligned}
$$
Similarly, iffirst two digits are $42,46,62$, or 64 , the remaining 4 digits can be arranged in same way i.e., 1680 ways.
$\therefore$ Total number of telephone numbers having all six digits
$$
\text { distinct }=5 \times 1680=8400 \text {. }
$$
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