Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ By expanding $( 1 + x ) ^ { n }$ for two appropriately chosen real numbers $x$, show that $$\sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } \binom { n } { 2 p } = 2 ^ { n - 1 }.$$

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^{m}\right)$ and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ are bases of $\mathbb{R}_{m}[X]$.

Show that:
$$( \cos ( t ) ) ^ { 2 p } = \frac { 1 } { 2 ^ { 2 p } } \left( \binom { 2 p } { p } + 2 \sum _ { k = 0 } ^ { p - 1 } \binom { 2 p } { k } \cos ( 2 ( p - k ) t ) \right)$$
Hint: One may develop $\left( \frac { e ^ { \mathrm{i} t } + e ^ { - \mathrm{i} t } } { 2 } \right) ^ { 2 p }$.

Show that, for all $n \in \mathbb { N } ^ { * }$,
$$\frac { 4 ^ { n } } { 2 n } \leqslant \binom { 2 n } { n } < 4 ^ { n } .$$

Show that, for all $t \in \mathbb{R}$, $$\sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{(t+h)x_i} = (2\operatorname{ch}(t+h))^n$$

155. The probability function is defined as $P(X = x) = \dfrac{\dbinom{5}{x}}{A}$\,;\; $x = 0, 1, 2, 3, 4, 5$. By calculating the value of $A$, what is $P(X = 2 \text{ or } X = 3)$?
(1) $\dfrac{3}{8}$ (2) $\dfrac{7}{16}$ (3) $\dfrac{9}{16}$ (4) $\dfrac{5}{8}$
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Prove that $2^n < \dbinom{2n}{n} < \dfrac{2^n}{\prod_{j=0}^{n-1}\left(1 - \frac{j}{n}\right)}$ for all positive integers $n$.

The value of $\lim_{n \to \infty} \frac{\sum_{r=0}^{n} {}^{2n}C_{2r} \times 3^{r}}{\sum_{r=0}^{n-1} {}^{2n}C_{2r+1} \times 3^{r}}$ is
(a) 0
(b) 1
(c) $\sqrt{3}$
(d) $(\sqrt{3}-1)/(\sqrt{3}+1)$

Using AM-GM inequality, find a lower bound for $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right)$, and determine which of the following holds:
(A) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) < \left(\dfrac{2^{10}}{11}\right)^{11}$
(B) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) = \left(\dfrac{2^{10}}{11}\right)^{11}$
(C) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) > \left(\dfrac{2^{10}}{11}\right)^{11}$
(D) None of the above

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

For $k \geq 1$, the value of $$\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$$ equals
(A) $\binom { n + k + 1 } { n + k }$
(B) $( n + k + 1 ) \binom { n + k } { n + 1 }$
(C) $\binom { n + k + 1 } { n + 1 }$
(D) $\binom { n + k + 1 } { n }$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is:
(a) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(b) $\frac { 6 ! } { 3 ! }$
(c) 33
(d) $3 \cdot \left( \frac { 6 ! } { 3 ! 3 ! } \right)$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is:
(a) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(b) $\frac { 6 ! } { 3 ! }$
(c) 33
(d) $3 \cdot \left( \frac { 6 ! } { 3 ! 3 ! } \right)$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is
(A) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$
(B) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

The number $$\left( \frac { 2 ^ { 10 } } { 11 } \right) ^ { 11 }$$ is
(A) strictly larger than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$
(B) strictly larger than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2$ but strictly smaller than $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$
(C) less than or equal to $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2$
(D) equal to $\binom{10}{1}^2 \binom{10}{2}^2 \binom{10}{3}^2 \binom{10}{4}^2 \binom{10}{5}$

The number $$\left( \frac { 2 ^ { 10 } } { 11 } \right) ^ { 11 }$$ is
(A) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(B) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$ but strictly smaller than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(C) less than or equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$
(D) equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$

Let the integers $a _ { i }$ for $0 \leq i \leq 54$ be defined by the equation
$$\left( 1 + X + X ^ { 2 } \right) ^ { 27 } = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { 54 } X ^ { 54 }$$
Then, $a _ { 0 } + a _ { 3 } + a _ { 6 } + a _ { 9 } + \cdots + a _ { 54 }$ equals
(A) $3 ^ { 26 }$
(B) $3 ^ { 27 }$
(C) $3 ^ { 28 }$
(D) $3 ^ { 29 }$.

The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3

Let $n$ be a positive integer and $t \in ( 0,1 )$. Then $\sum _ { r = 0 } ^ { n } r \binom { n } { r } t ^ { r } ( 1 - t ) ^ { n - r }$ equals
(A) $n t$
(B) $( n - 1 ) ( 1 - t )$
(C) $n t + ( n - 1 ) ( 1 - t )$
(D) $\left( n ^ { 2 } - 2 n + 2 \right) t$.

The value of $$\sum _ { k = 0 } ^ { 202 } ( - 1 ) ^ { k } \binom { 202 } { k } \cos \left( \frac { k \pi } { 3 } \right)$$ equals
(A) $\quad \sin \left( \frac { 202 } { 3 } \pi \right)$.
(B) $- \sin \left( \frac { 202 } { 3 } \pi \right)$.
(C) $\quad \cos \left( \frac { 202 } { 3 } \pi \right)$.
(D) $\cos ^ { 202 } \left( \frac { \pi } { 3 } \right)$.

Let $n > 1$, and let us arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the first three coefficients are in arithmetic progression. Then, the number of terms where $x$ appears with an integer power, is
(A) 3
(B) 2
(C) 1
(D) 0

For $k \geq 1$, the value of $\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$ equals
(a) $\binom { n + k + 1 } { n + k }$.
(B) $( n + k + 1 ) \binom { n + k } { n + 1 }$.
(C) $\binom { n + k + 1 } { n + 1 }$.
(D) $\binom { n + k + 1 } { n }$.

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. }

22. If in the expansion of $( 1 + x ) n$, the coefficients of $x$ and $x 2$ are 3 and - 6 respectively, then m is :
(A) 6
(B) 9
(C) 12
(D) 24