1 . Let $D_1,\ldots,D_{2n}$ be the result of rolling a $6$-sided die $2n$ times, and consider the length-$n$ sequence
\[ S=\langle (D_1+D_2), (D_3+D_4), (D_5+D_6),\ldots, (D_{2n-1}+D_n)\rangle \]
whose entries are all in the set $\{2,3,\ldots,12\}$. Let $k$ be an integer in $\{0,\ldots,n\}$. What is the probability that $S$ contains exactly $k$ many $5$'s?
(a)
$\binom{n}{k}\cdot(\tfrac{1}{36})^k\cdot (\tfrac{35}{36})^{n-k}$
(b)
$\binom{n}{k}\cdot(\tfrac{1}{11})^k\cdot (\tfrac{10}{11})^{n-k}$
(c)
$\binom{n}{k}\cdot(\tfrac{1}{12})^k\cdot (\tfrac{11}{12})^{n-k}$
(d)
$\binom{n}{k}\cdot(\tfrac{1}{9})^k\cdot (\tfrac{8}{9})^{n-k}$
(e)
$\binom{n}{k}\cdot(\tfrac{1}{5})^k\cdot (\tfrac{4}{5})^{n-k}$