8. Leonardo Sinisgalli writes, in a passage from Furor Mathematicus: ``I had in mind a chapter on the laws of chance: I wanted to find the relationships between Tartaglia's triangle, relating to the coefficients of the polynomial $( a + b ) ^ { n }$ and Pascal's arithmetic triangle, which gives us the probability of getting $m$ tails in $n$ games played at heads and tails''. Describe the relationship existing between binomial coefficients and the calculation of probabilities.
\footnotetext{Maximum duration of the exam: 6 hours. The use of scientific or graphical calculators is allowed provided they are not equipped with symbolic algebraic processing capability and do not have Internet connectivity. The use of a bilingual dictionary (Italian–language of the country of origin) is allowed for candidates whose native language is not Italian. It is not allowed to leave the Institute before 3 hours have elapsed from the delivery of the exam text. }