A set $S$ contains $(2n+1)$ elements. If the number of subsets of $S$ which contain at most $n$ elements is $1024$, then what is the value of $n$ ?

The number of elements in the set $S$ is $2n+1$.

The number of subsets of $S$ containing at most $n$ elements is given by the sum:

$\sum_{k=0}^{n} {}^{2n+1}C_{k}$

We know the sum of all binomial coefficients is:

$\sum_{k=0}^{2n+1} {}^{2n+1}C_{k} = 2^{2n+1}$

Using the symmetry property of binomial coefficients, $^{n}C_{r} = ^{n}C_{n-r}$, we have:

$\sum_{k=0}^{n} {}^{2n+1}C_{k} = \sum_{k=n+1}^{2n+1} {}^{2n+1}C_{k}$

Therefore, the sum of the first half of the coefficients is exactly half of the total sum:

$\sum_{k=0}^{n} {}^{2n+1}C_{k} = \dfrac{2^{2n+1}}{2} = 2^{2n}$

Given that the number of such subsets is $1024$, we can write:

$2^{2n} = 1024$

Since $1024 = 2^{10}$, we get:

$2n = 10$

$n = 5$

Answer: $5$