JEE Main 2023 — Mathematics Algebra

Given,

We have to form $6$6 digit number using the digits 4,5&9 and for number should be divisible by $6$6 which means that the addition of number should be divisible by $3$3 and unit place should be even, so we will make following cases:

Case $(1)$(1) The number is $444444$444444 which can be arranged in $1$1 way,

Case $(2)$(2) when the number is formed using the digits $4,4,4,5,9$4,4,4,5,9 and last place is fixed with $4$4, that can be arranged in $\frac{5!}{3!}=20\text{ways}$3!5!​=20ways

Case $(3)$(3) when number is formed using the digits $4,4,5,5,5$4,4,5,5,5 and last place is fixed with $4$4, that can be arranged in $\frac{5!}{3!2!}=10\text{ways}$3!2!5!​=10ways

Case $(4)$(4) when number is formed using the digits $4,4,9,9,9$4,4,9,9,9 and last place is fixed by $4$4, that can be arranged in $\frac{5!}{3!2!}=10\text{ways}$3!2!5!​=10ways

Case $(5)$(5) when number is formed using the digits $4,5,5,9,9$4,5,5,9,9 and last place is fixed with $4$4, that can be arranged in $\frac{5!}{2!2!}=30\text{ways}$2!2!5!​=30ways

Case $(6)$(6) when number is formed using the digits $5,9,9,9,9$5,9,9,9,9 and last place is fixed with $4$4, that can be arranged in $\frac{5!}{4!}=5\text{ways}$4!5!​=5ways

Case $(7)$(7) when number is formed using the digits $5,5,5,5,9$5,5,5,5,9 and last place is fixed with $4$4, that can be arranged in $\frac{5!}{4!}=5\text{ways}$4!5!​=5ways

Now adding all the cases we get, $1+20+10+10+30+5+5=81\text{ways}$1+20+10+10+30+5+5=81ways.