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Q.
Number of numbers greater than a million and divisible by $5$ which can be formed by using only the digits $1,2,1,2,0,5 \& 2$ is -
Permutations and Combinations
Solution:
For a number to be divisible by $5,5$ or $0$ should be at units place.
$\therefore $ Unit place can be filled by $2$ ways
Remaining digits can be filled in $\frac{6 !}{3 ! \times 2 !}$ ways.
$\therefore$ Total ways $=\frac{2 \times 6 !}{3 ! 2 !}$
But these arrangements also include cases where 0 is at millions place and 5 at units place, which are undesirable cases
$\Rightarrow \frac{5 !}{3 ! \times 2 !}$ ways (undesirable)
subtract it from total ways.
$\therefore$ Desired ways $=2 \frac{6 !}{3 ! \times 2 !}-\frac{5 !}{3 ! \times 2 !}=110$