a) In a survey on the ice-cream preferences towards college students, the following data was obtained:70 PI 78 like strawberry 32 like chocolate 57 like tiramisu 13 like both strawberry and chocolate 21 like chocolate and tiramisu 16 like both tiramisu and strawberry 5 like all three flavors 14 like none of these three flavors How many students were surveyed? b) Prove the following Pascal's theorem. \text { Let } n \text { and } r \text { be positive integers and suppose } r \leq n \text {. Then }\left(\begin{array}{c}

n+1 \\

r

\end{array}\right)=\left(\begin{array}{c}

n \\

r-1

\end{array}\right)+\left(\begin{array}{c}

n \\

r

\end{array}\right) \text {. } Expand the following expressions using Binomial Theorem: \text { i. }(a+b)^{5} \text { ii. }(x-2 y)^{4} \text { iii. }\left(p^{2}-2 q\right)^{3} d) By using the pigeonhole principle, explain how any set of 76 positive integers that less than or equal to 100 must contains 4 consecutive integers.