To determine how many cups of punch Lani makes, we need to add the amounts of each ingredient together. Here's the expression using part-whole reasoning:

1. Orange juice: $5 \frac{3}{4}$ cups
2. Pineapple juice: $3 \frac{1}{3}$ cups
3. Seltzer water: $2 \frac{5}{6}$ cups

The total amount of punch Lani makes is the sum of these three amounts. The expression is:

$5 \frac{3}{4} + 3 \frac{1}{3} + 2 \frac{5}{6}$

Now, let's add them step by step:

1. Convert mixed numbers to improper fractions:

   $5 \frac{3}{4} = \frac{4 \times 5 + 3}{4} = \frac{23}{4}$ $3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$ $2 \frac{5}{6} = \frac{6 \times 2 + 5}{6} = \frac{17}{6}$

2. Find a common denominator and add the fractions:

   The least common denominator for 4, 3, and 6 is 12. $\frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12}$ $\frac{10}{3} = \frac{10 \times 4}{3 \times 4} = \frac{40}{12}$ $\frac{17}{6} = \frac{17 \times 2}{6 \times 2} = \frac{34}{12}$

3. Add the fractions: $\frac{69}{12} + \frac{40}{12} + \frac{34}{12} = \frac{143}{12}$

4. Convert the improper fraction back to a mixed number: $\frac{143}{12} = 11 \frac{11}{12} \text{ cups}$

Thus, Lani makes a total of $11 \frac{11}{12}$ cups of punch.

Would you like more details on any part of this solution?

Here are some related questions:

1. What if Lani used $4 \frac{1}{2}$ cups of seltzer water instead?
2. How much punch would Lani make if she doubled the amount of orange juice?
3. What if the pineapple juice was increased by $\frac{2}{3}$ cup? How much punch would she make then?
4. How many cups of punch would Lani make if she used $1 \frac{1}{4}$ cups of cranberry juice in addition?
5. What if Lani used 1 cup less of orange juice? How much punch would she then have?

Tip: When adding mixed numbers, it can be easier to first convert them to improper fractions, especially when working with different denominators.