The relative lack of students studying post-compulsory STEM (Science, Technology, Engineering and Mathematics) subjects is a key policy concern. A particular issue is the disparities in uptake by students' family background, gender and ethnicity. It remains unclear whether the effects of family background can be explained by academic disparities, and prior research has only considered student characteristics as having additive effects, and has not yet considered whether these characteristics interact in determining choices. Using data from more than 4,000 students in England from the "Next Steps", I use logistic regression methods to consider these questions. Disparities by students' ethnicity are shown to increase when considering prior attainment. Parents' education and social class are differentially related to uptake for male and female students. Disparities in degree choice by parents' education persist when conditioning on prior attainment suggesting a leak in the pipeline', particularly for young women from less educated families. Disparities in uptake by gender are only observed for low and middle Socio-Economic Position (SEP) families. Implications for policy and future research are discussed.