Let the original number of boys be \\(b\\) and girls be \\(g\\).

After enrolment: boys = \\(1.10b\\), girls = \\(1.04g\\).
Ratio: \\(1.10b : 1.04g = 11 : 12\\).
\\(1.10b \times 12 = 1.04g \times 11\\)
\\(13.2b = 11.44g\\)
\\(b = \frac{13}{15}g\\)

New pupils: \\(0.10b + 0.04g = 17\\)
\\(0.10(\frac{13}{15}g) + 0.04g = 17\\)
\\(\frac{1.3}{15}g + \frac{0.6}{15}g = 17\\)
\\(\frac{1.9}{15}g = 17\\)
\\(g = \frac{2550}{19} \approx 134.21\\)
\\(b = \frac{13}{15} \times \frac{2550}{19} = \frac{2210}{19} \approx 116.32\\)

Total at first = \\(\frac{4760}{19} \approx 250.53\\)

Final Answer: \\(\frac{4760}{19}\\) (approximately 251)