How do I earn the parent's trust?

I found a job that requires me to babysit full time for 3 school aged girls (8, 9 and 10). Basically, the mom can't afford a camp for 3 kids so she'd like me to keep them happy and entertained during the day. Since they're a little far, the babysitting would take place at my house. I am a little nervous about this because I am afraid there won't be enough space for them (my family of 5 lives in an apartment with 2 bedrooms, 2 bathrooms, 2 balconies, a kitchen and a large living but they only have access to 1 bedroom, the living room, a bathroom, a balcony and the kitchen). Our house is clean, but we don't have a lot of toys/electronics. On the other side, we have lots of libraries, swimming pools, parks and playgrounds surrounding our community. Our neighbourhood is amazing, it's safe and quiet. The mom would like the girls to meet me a few times and play with me before I start babysitting. I really want this job, but I am nervous. What games should we play and how do I prepare before they come to our house? How do I earn the mom's trust?