Question: You are given $20$ sugar bars (B$_{1}$, B$_{2}$, ..., B$_{20}$) and $50$ salt bars (S$_{1}$, S$_{2}$, ..., S$_{50}$). Consider subsets of these $70$ bars, that contain at least $3$ sugar bars (and any number of salt bars). How many such subsets are there?
(Answer: $1.18 \times 10^{21}$)
Attempt: Since it is at least $3$ sugar bars, it should start the count at $\dbinom{20}{3}$ all the way to $\dbinom{20}{20}$. For the salt bars, since it is any number of combinations, it should be $2^{50}$. I just multiplied them out using product rule but I didn't get the correct answer. How should I proceed with this?
Total number of subsets is $2^{70}$. consider those that have less than 3 sugar bars, $\binom{20}{2}2^{50}+\binom{20}{1}2^{50}+2^{50}$. so we should get $2^{70}-\binom{20}{2}2^{50}-\binom{20}{1}2^{50}-2^{50}$. I hope I got this right!