Let's identify the missing combinations of three different scoops of gelato from the following list:
1. Hazelnut (H)
2. Chocolate (C)
3. Tiramisu (T)
4. Pistachio (P)
5. Strawberry (S)
First, we list all possible combinations of choosing 3 scoops out of 5 flavors. There are combinatorial ways to do this. Specifically, the number of combinations is calculated by the combination formula [tex]\( \binom{n}{k} \)[/tex] , where [tex]\( n \)[/tex] is the total number of items and [tex]\( k \)[/tex] is the number of items to choose. Here, [tex]\( n = 5 \)[/tex] and [tex]\( k = 3 \)[/tex]:
[tex]\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \][/tex]
So, there are 10 possible combinations. Let's list them:
1. H - C - T
2. H - C - S
3. H - T - S
4. C - T - P
5. C - P - S
6. H - C - P
7. H - P - S
8. T - P - S
9. H - T - P
10. C - T - S
Next, we cross-check each of these combinations with the given list to find the missing ones:
| Given List | Found in Complete List? |
|-------------------------|-------------------------|
| H - C - T | Yes |
| H - C - S | Yes |
| H - T - S | Yes |
| C - T - P | Yes |
| C - P - S | Yes |
| H - C - P | Yes |
| ? (First missing) | No |
| H - P - S | Yes |
| ? (Second missing) | No |
| T - P - S | Yes |
By comparing the lists, we determine that the missing combinations in the table are:
For the first missing:
- H - T - P
For the second missing:
- C - T - S
Hence, the complete list with missing combinations filled in would be:
[tex]\[
\begin{tabular}{|ccccc|}
\hline H - C - T & H - C - S & H - T - S & C-T-P & C-P-S \\
H - C - P & H - T - P & H - P - S & C - T - S & T-P-S \\
\hline
\end{tabular}
\][/tex]
The two missing answers from the given options are:
A. H - T - P
B. C - T - S
