Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Establish, for every non-zero natural number $n$, that $Y_n$ follows a uniform distribution on $D_n$.

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Establish, for every non-zero natural number $n$, that $Y_n$ follows a uniform distribution on $D_n$.

In a factory, 20 workers start working on a project of packing consignments. They need exactly 5 hours to pack one consignment. Every hour 4 new workers join the existing workforce. It is mandatory to relieve a worker after 10 hours. Then the number of consignments that would be packed in the initial 113 hours is
(A) 40
(B) 50
(C) 45
(D) 52.