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In this question , I dont understand why the KBC is 4EI / L .... I was told that for the far end pinned or roller supported , the K = 3EI / L , so shouldn't the KBC = 3EI / L

In the 3rd photo , we can see that for far end pinned supported , k = 3EI / L , not 4EI / L

What are the difference between the example in photo 2 and photo 3 ? gg

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The far end formula, $\frac{3EI}{L}$, applies if the beam is discontinuous in that point. This means that the point of reference is the end of the beam where the support is a hinge/pin.

Example: $K_{AB}$. Note the $A$ in the $AB$. Point $A$ is the reference point.

The pin support in picture two is not discontinuous in point B, therefore, the distribution factor is still $\frac{4EI}{L}$.

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