How many ways can the digits $2,3,4,5,6$ be arranged to get a number divisible by 11

How many ways can the digits $2,3,4,5,6$ be arranged to get a number divisible by $11$

I know that the sum of the permutations of the digits should be divisible by 11. Also, the total number of ways the digits can be arranged is $5! = 120$.

• Since the digits add up to an even number, the alternating sum of the digits has to be a multiple of $22$.The only way to get that is to get $0$ for the alternating sum. – Thomas Andrews Feb 6 '18 at 4:54
• 24365 = 11*2215 is divisible by 11. – Steve Kass Feb 6 '18 at 5:00

Hint. By the divisibility rule by $11$ we have to count the arrangements $d_1,d_2,d_3,d_4,d_5$ of the digits $2,3,4,5,6$ such that $d_1+d_3+d_5-(d_2+d_4)$ is divisible by $11$. Notice that $$-2=2+3+4-(5+6)\leq d_1+d_3+d_5-(d_2+d_4)\leq 4+5+6-(2+3)=10$$ therefore we should have $d_1+d_3+d_5=d_2+d_4=\frac{2+3+4+5+6}{2}=10$.

In how many ways we can do that?

Let the base-$10$ number $abcde$ be divisible by $11,$ where $\{a,b,c,d,e\}=\{2,3,4,5,6\}.$ The base-$10$ number $abcde$ is divisible by $11$ iff $11$ divides $(a+c+e)-(b+d).$

We cannot have $a+c+e\geq 11 +b+d$ because $a+c+e\leq 4+5+6=15$ and $11+b+d\geq 11+2+3=16.$

We cannot have $b+d\geq 11+a+c+e$ because $b+d\leq 5+6=11$ and $11+a+c+e\geq 11+2+3+4=20.$

So the only multiple of $11$ that can be equal to $(a+c+e)-(b+d)$ is $0.$ So $a+c+e=b+d.$ Hence $2(b+d)=(a+c+e)+(b+d)=20.$

So $b+d=10.$

So $6\in \{b,d\}.$ Otherwise $b+d\leq 5+4=9$, but $b+d=10$. Since $6\in \{b,d\}$ and $b+d=10$ we have $\{b,d\}=\{6,4\}.$ Hence $\{a,c,e\}=\{2,3,5\}.$

There are $3!$ permutations of $(2,3,5)$ and $2!$ permutations of $(4,6)$ so there are $3!\cdot 2!=12$ ways.

The total sum of all digits given is $20$. Hence we need to break it into two equal parts which is $10$ each. And the only pair forming it using given digits is {2,3,5}, {4,6}

We know the three digits can be placed only on $1^{st},3^{rd}$ and $5^{th}$ places respectively while the rest on remaining. Hence the number of ways is simply permitting 3 numbers and 2 numbers at the same time.

Hence the answer would be simply $3!*2!=12$