# How many subsets are there for this case?

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?

• On this site, we use a type of $\LaTeX$ called MathJax. Search for a tutorial on it. Please use it in future :) – Shaun Nov 25 '18 at 4:10

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!