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The number of four digit numbers that can be formed using the digits $1,2,3,4,5,6,7$ which are divisible by 4 , when the repetition of any digit is not allowed, is
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The correct answer is:
$200$
$1,2,3,4,5,6,7$
For number of numbers divisible by 4 , unit digit can be either of $2,4,6$.
When unit digit is 2 , digit at tens place can be $1,3,5,7$.
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{4} \times \frac{2}{1}=80$
When unit digit is 4 , digits at tens place can be 2,6 .
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{2} \times \frac{4}{1}=40$
When unit digit is 6 , digit at tens place can be $1,3,5,7$.
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{4} \times \frac{6}{1}=80$
$\therefore \quad$ Total number of numbers $=80+40+80=200$.
For number of numbers divisible by 4 , unit digit can be either of $2,4,6$.
When unit digit is 2 , digit at tens place can be $1,3,5,7$.
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{4} \times \frac{2}{1}=80$
When unit digit is 4 , digits at tens place can be 2,6 .
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{2} \times \frac{4}{1}=40$
When unit digit is 6 , digit at tens place can be $1,3,5,7$.
$\therefore \quad$ Possible numbers $\frac{}{4} \times \frac{}{5} \times \frac{}{4} \times \frac{6}{1}=80$
$\therefore \quad$ Total number of numbers $=80+40+80=200$.
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