Find the number of n digit-numbers formed using the first 5 natural numbers, that contain the digits '2' and '4', essentially.
I tried attempting this with the inclusion- exclusion principle but got stuck with the first 5 natural numbers condition. I made 2 sets, one with natural numbers where 2 is there and another with numbers with 5 is there and tried to find their intersection, but got stuck while subtracting from the total set, ie, the set of natural n digit numbers that are made from the digits 1 to 5 (wo any restrictions) hence i have a doubt in how to find the total number of elements in the set of n digit numbers that are made from the first 5 natural numbers.
Can someone please help me with this?
The total number of $n$-digit numbers formed from $12345$ is $5^n$. The count of those numbers not containing a $2$ or $4$ is $4^n$ each, and the count of those not containing either is $3^n$. Thus the required answer is $5^n-2\cdot4^n+3^n$.