The problem you're asking involves a summation of binomial coefficients and trigonometric terms, which seems complex. To solve this, let's break down the given expression step-by-step.

The expression for $S$ is given as:

$S = \sum_{r=0}^{n} \frac{\binom{n}{r}}{\sum_{s=0}^{r} \binom{r}{s} \sum_{t=0}^{s} \binom{s}{t} \left( 2 \sin \frac{\pi}{10} \right)^t}$

You are tasked with finding $S^{1/n}$.

Solution Strategy

1. Break down the summation:
   The upper summation for $r$ and the nested binomial coefficients with trigonometric terms seem to imply a recursive application of binomial expansions. This resembles expansions related to trigonometric identities involving binomial coefficients.

2. Pattern Recognition:
   The term $\left( 2 \sin \frac{\pi}{10} \right)^t$ inside the summation suggests that this is related to multiple angle identities for sine or cosine.

Simplification:

The expression appears to be related to summations leading to trigonometric constants. From binomial expansions of trigonometric functions, you could anticipate that the result of this expression leads to a standard form.

The Correct Answer:

By recognizing the pattern and standard results of such binomial expansions, the answer simplifies to:

$S^{1/n} = 2 \cos \frac{\pi}{10}$

Final Answer:

The correct option is:

$\boxed{b} \ 2 \cos \frac{\pi}{10}$

Would you like more detailed insights into binomial expansions and how they relate to trigonometric identities?

Here are some related questions to expand on this:

1. How are binomial expansions used in deriving trigonometric identities?
2. Can you provide more examples of binomial summations involving sine or cosine terms?
3. What is the connection between binomial coefficients and multiple angle identities in trigonometry?
4. How would this problem change if the angle inside the sine function were different?
5. How can one systematically approach nested summations like the one in this problem?

Tip: Binomial summations often relate to generating functions, which can simplify complex summations into known trigonometric or exponential forms.