We can start by using the formula for the average of a set of numbers:

average = \frac{\text{sum of numbers}}{\text{number of numbers}}

We know that the average number of pages in the first five books is 78, so we can write:

78 = \frac{\text{sum of pages in the first five books}}{5}

Multiplying both sides by 5, we get:

\text{sum of pages in the first five books} = 390

We also know that adding N books with an average of P pages per book increases the overall average by 2 pages per book. This means that the new average is 78 + 2 = 80, so we can write:

80 = \frac{\text{sum of pages in all } N+5 \text{ books}}{N+5}

Multiplying both sides by N+5, we get:

\text{sum of pages in all } N+5 \text{ books} = 80(N+5)

We can now set up an equation to solve for N. We know that the sum of pages in the first five books is 390, and the sum of pages in all N+5 books is 80(N+5), so we can write:

390 + N * P = 80(N + 5)

Simplifying this equation, we get:

390 + N * P = 80N + 400

Subtracting 80N from both sides, we get:

N * P - 80N = 10

Factoring out N, we get:

N(P - 80) = 10

Dividing both sides by (P - 80), we get:

N = \frac{10}{P - 80}

Therefore, the answer is:

N = \frac{10}{P-80}

So the correct option is \mathbf{A.}\ N = 10:(p-80).