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What is The commonest hip injury in the elderly patients is - and explain in detail?
First see statements of different textbooks:-

"The femoral neck is the commonest site of fracture in the elderly".   —Apley's
"Intertrochanteric fracture is four times more common than neck fracture".  —Ebnezar
"More-over fracture intertrochanteric femur occurs in elderly patients even more than fractures of the femoral neck itself". — Watson Jones
"Femoral neck fractures and intertrochanteric fractures occur with about the same frequency".— Campbell

I cannot comment about the commonest fracture among these two.
However, I am sure about the answer as extracapsular fracture includes both extracapsular neck fracture (Basal / Basicervical fracture) and inter-trochanteric fracture. Basal (Basicervical) fracture is the most common type of neck fracture. So, even if neck fracture and IT fracture occur with same frequency, extracapsular fractures (Basal fracture + Intertrochanteric fracture) are more common than intracapsular (transcervical and subcapital) fractures.