Step 1: Identify the pattern. Urine that escapes all the time, with the woman unable to pass urine voluntarily, signals a leak between the urinary tract and the vagina rather than a storage or sphincter problem. The timing, about a week after hysterectomy, fits a surgical fistula.
Step 2: Test each option against the anatomy left after surgery. The uterus has been taken out, so no vesicouterine fistula can form, eliminating option (c) on logical grounds alone. A urethrovaginal fistula tends to leak only while voiding rather than continuously, so it does not match constant dribbling.
Step 3: That narrows it to a bladder or ureteric injury. The bladder sits directly in front of the cervix and uterus and is the organ most frequently injured at hysterectomy. A direct bladder-to-vagina opening produces ceaseless leakage with no voluntary control, which is exactly the described scenario. A ureterovaginal fistula usually still lets the bladder fill, so intermittent voluntary voiding persists, making it a less complete fit.
Step 4: The constant involuntary urine loss after hysterectomy is therefore a vesicovaginal fistula.
\[\boxed{\text{Vesicovaginal fistula}}\]