$(2)$ More inclusion$-$exclusion counting: How many bit strings of length $15$ have bits $1,2,$ and $3$ equal to $101,$ or have bits $12,13,14,$ and $15$ equal to $1001$ or have bits $3,4,5,$ and $6$ equal to $1010?($Number bits from left to right. In other words, bit #$1$ is the left most bit and bit #$15$ is the right most bit.$)$

Hint: The fact that the third bit appears in two of the required patterns means some special care will be needed to get the count correct.