Suppose an $8$-digit number will be formed using the digits from $1$ to $9$. Find the cardinality when exactly $5$ consecutive digits are even and exactly $2$ digits are odd, e.g., $18242674$, $26682523$
I thought of dividing it into four subgroups, the five consecutive even integers, the two odd numbers, and the other even integer. ie, $(4!) (4)^5 \times (5)^2 \times (4)^1$. I plan to use this cardinality for the classical approach of a prob. Thanks.