Question

The number of $6$ digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by $11$ and no digit is repeated, is :

• A. $48$
• B. $60$
• C. $72$
• D. $36$

Answer 1

The correct option is B $60$

For a number to be divisible by $11$, difference of sum of alternate digits of the number should be of the form $11n$ where $n\in \{0,1,2,\mathrm{3....}\}$
Let the digits be $a,b,c,d,e,f$
$|a+c+e-(b+d+f)|=11n$

Therefore, only possible combination is
$\{a,c,e\},\{b,d,f\}=\{9,2,1\},\{7,5,0\}$
Therefore, required no. of ways $=2\cdot 3!\cdot 3!-2!\cdot 3!$ $(\because 0$ cannot be the first digit$)$
$\Rightarrow 72-12=60$