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. }

Q63. If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$ , then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5

For every subsets $A$ and $B$ of a non-empty set $X$, the operation $\odot$ is defined as
$$\mathrm { A } \odot \mathrm { B} = \mathrm { X } \backslash ( \mathrm { A} \cup \mathrm { B} )$$
For every subsets $K$ and $L$ of X satisfying the condition $K \subseteq L$, what is the result of the operation $$( \mathbf { X } \backslash \mathbf { L } ) \odot ( \mathbf { L } \backslash \mathbf { K } )$$
A) $X$
B) K
C) L
D) $X \backslash K$
E) $X \backslash L$