Do all B777 family aircraft have the same engines?
TL;DR: No.



For the true answer refer to the FAA Type Certificate Data Sheet

777-200

2 Pratt and Whitney Turbofan Model:
PW4074, PW4074D, PW4077, PW4077D,PW4090,PW4084D, and PW4090-3
(Engine Type Certificate No. E46NE)

2 General Electric Turbofan Model: GE90-76B, GE90-85B, GE90-90B, GE90-94B
(Engine Type Certificate No. E00049EN)

2 Rolls-Royce Turbofan Model: RB211-Trent 875-17, RB211-Trent, 877-17,
RB211-Trent 884-17, RB211-Trent 892-17, or RB211 Trent 892B-17, RB211 Trent 895-17
(Engine Type Certificate E00050EN) 

777-300

2 Rolls-Royce Turbofan Model: RB211-Trent 884-17, RB211-Trent 884B-17,
or RB211-Trent 892-17
(Engine Type Certificate E00050EN)

2 Pratt &amp; Whitney Turbofan Model: PW4090, PW4098
(Engine Type Certificate E46N)

777-300ER

2 General Electric Turbofan Model: GE90-115B
(Engine Type Certificate No. E00049EN) 

777-200LR

2 General Electric Turbofan Model: GE90-110B1

2 General Electric Turbofan Model: GE90-115B

(Engine Type Certificate No. E00049EN) 

777F

2 General Electric Turbofan Model: GE90-110B1

2 General Electric Turbofan Model: GE90-115B
(Engine Type Certificate No. E00049EN)