1. Given data

• Wavelength of light: $\lambda = 5303\,\text{\AA} = 5303 \times 10^{-10}\,\text{m} = 5.303 \times 10^{-7}\,\text{m}$
• Slit separation: $d = 19.44\,\mu\text{m} = 19.44 \times 10^{-6}\,\text{m}$
• Slit width: $a = 4.05\,\mu\text{m} = 4.05 \times 10^{-6}\,\text{m}$

We need the number of interference bright fringes between the first and second diffraction minima.

2. Conditions for interference maxima and diffraction minima

• Interference bright fringes occur at: $d\sin\theta = n\lambda$ where $n = 0, \pm 1, \pm 2, \dots$

• Diffraction minima occur at: $a\sin\theta = m\lambda$ where $m = 1, 2, 3, \dots$

The first and second diffraction minima on one side correspond to: $a\sin\theta = \lambda \quad \text{and} \quad a\sin\theta = 2\lambda$

So the bright fringes lying between them must satisfy: $\lambda < a\sin\theta < 2\lambda$

Using $d\sin\theta = n\lambda$, $a\cdot \frac{n\lambda}{d} \in (\lambda, 2\lambda)$

Cancelling $\lambda$: $1 < \frac{an}{d} < 2$

Thus, $\frac{d}{a} < n < \frac{2d}{a}$

3. Compute $\dfrac{d}{a}$

$\frac{d}{a} = \frac{19.44}{4.05} = 4.8$

Therefore, $4.8 < n < 9.6$

The integer values of $n$ satisfying this are: $n = 5,6,7,8,9$

So the number of bright fringes is: $5$

4. Check with options

The correct option is: $\boxed{\text{B: }05}$

5. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They match.