Question

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _______.

Answer 1

Answer: 576

Digits are $1,2,3,4,5,7,9$
Multiple of $11\to$ Difference of sum at even $\&$ odd place is divisible by $11$ .
Let number of the form $abcdefg$
$\therefore (a+c+e+g)-(b+d+f)=11x$
$a+b+c+d+e+f=31$
$\therefore$ either $a+c+e+g=21$ or 10
$\therefore b+d+f=10$ or 21
Case- $1$
$a+c+e+g=21$
$b+d+f=10$
(b, d, f) $\in \{(1,2,7)(2,3,5)(1,4,5)\}$
(a, c, e, g) $\in \{(1,4,7,9),(3,4,5,9),(2,3,7,9)\}$
$\therefore$ Total number in case- $1=(3!\times 3)(4!)=432$
Case- 2
$a+c+e+g=10$
$b+d+f=21$
$(a,b,e,g)\in \{1,2,3,4)\}$
$(b,d,f)\&\{(5,7,9)\}$
$\therefore$ Total number in case $2=3!\times 4!=144$
$\therefore$ Total numbers $=144+432=576$