The problem asks to find the number of subsets of size 5 for the set $H = \{ c, e, r, d, a, s \}$.

Solution:

The formula to determine the number of subsets with $r$ elements from a set with $n$ elements is:

$\binom{n}{r} = \frac{n!}{r! \cdot (n-r)!}$

Here:

• $n = 6$ (the number of elements in $H$),
• $r = 5$ (the size of the subset).

Substituting into the formula:

$\binom{6}{5} = \frac{6!}{5! \cdot (6-5)!} = \frac{6!}{5! \cdot 1!}$

Simplify:

$\binom{6}{5} = \frac{6 \cdot 5!}{5! \cdot 1} = 6$

Thus, the number of subsets of $H$ with 5 elements is 6.

Final Answer:

The correct option is not visible in the image, but the number of subsets with 5 elements is 6.

Let me know if you'd like more explanation or have further questions! Here are five related questions for you to consider:

1. How do you calculate subsets of a different size from the same set?
2. What is the total number of subsets of a set with $n$ elements?
3. Can you explain the difference between combinations and permutations?
4. How would the formula change if the order of elements in the subsets mattered?
5. What is the significance of factorials in combination calculations?

Tip: When solving combination problems, always double-check if order matters—this determines whether to use permutations or combinations!