Likely fake, based on:
• uniformity of size, surface appearance and color tone of pearls
• the low end way it is finished (the bead tips that connect the necklace to the clasp)
• large knots (fake pearls tend to have larger drill holes than real pearls) and thicker thread makes larger knots)
• if the clasp is base metal the pearls are more likely to be fake
Try this simple test: Rub two of the pearls against each other or rub one against the edge of your front tooth (after wiping them clean with a damp soft cloth). Real pearls would feel a bit gritty; fake pearls would glide smoothly.
Even if fake, there may be sentimental value in the pearls anyway. That generation often wore fake pearls. My husband's late mother, as well as mine, wore fake pearls.