Question

How many six-digit numbers are there in which no digit is repeated, even digits appear at even places,odd digits appear at odd places and the number is divisible by $4$ ?

• A. $3600$
• B. $2700$
• C. $2160$
• D. $1440$

Answer 1

Answer: (D)

$1440$

We need to fill the above spaces in such a way that places $2,4,6$ have even numbers and the rest have odd. Also, none of the numbers must be repeated. Since the number is divisible by $4$, the possible numbers for $5$ and $6$ are $6$ can be either $2,6$ and $5$ can be an odd number.

$\Rightarrow$ Combinations $={}^2{C}_1\times {}^5{C}_1\times {}^4{C}_1\times {}^4{C}_1\times {}^3{C}_1\times {}^3{C}_1$

$6$ $5$ $4$ $3$ $2$ $1$

$=2\times 5\times 16\times 9$

$\Rightarrow$ Total $=1440$

Hence, the answer is $1440.$