How many numbers formed using the digits 3,2,0,4,3,2,3 exceed one… - Vidyadip

Question

How many numbers formed using the digits $3,2,0,4,3,2,3$ exceed one million?

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Answer

A number that exceeds one million is to be formed from the digits $3,2,0,4$, $3,2,3$.
Then the numbers should be any number of 7 digits which can be formed from these digits.
Also among the given numbers 2 repeats twice and 3 repeats thrice.
$\therefore$ Required number of numbers $=$ Total number of arrangements possible among these digits - number of arrangements of 7 digits which begin with 0.
$
\begin{aligned}
& =\frac{7 !}{2 ! 3 !}-\frac{6 !}{2 ! 3 !} \\
& =\frac{7 \times 6 \times 5 \times 4 \times 3 !}{2 \times 3 !}-\frac{6 \times 5 \times 4 \times 3 !}{2 \times 3 !} \\
& =7 \times 6 \times 5 \times 2-6 \times 5 \times 2 \\
& =6 \times 5 \times 2(7-1) \\
& =60 \times 6 \\
& =360
\end{aligned}
$
$\therefore 360$ numbers that exceed one million can be formed with the digits $3,2,0$, $4,3,2,3$.

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