Let's solve this problem step by step.

Given:

• $S = \{15, 16, 17, 18, 19\}$
• $T = \{8, 9, 10, 11, 12, 13, 14\}$

Task:

You need to choose one number from $S$ and one from $T$, then add them up. The question asks how many different sums can be obtained.

Step 1: Determine the Range of Possible Sums

• The smallest sum occurs when you add the smallest numbers from each set: $15 + 8 = 23$.
• The largest sum occurs when you add the largest numbers from each set: $19 + 14 = 33$.

So, the possible sums range from 23 to 33.

Step 2: Identify All Possible Sums

Now, let's calculate all the possible sums:

For each element in $S$, add it to each element in $T$:

• 15 + T: $15 + 8 = 23$, $15 + 9 = 24$, $15 + 10 = 25$, $15 + 11 = 26$, $15 + 12 = 27$, $15 + 13 = 28$, $15 + 14 = 29$
• 16 + T: $16 + 8 = 24$, $16 + 9 = 25$, $16 + 10 = 26$, $16 + 11 = 27$, $16 + 12 = 28$, $16 + 13 = 29$, $16 + 14 = 30$
• 17 + T: $17 + 8 = 25$, $17 + 9 = 26$, $17 + 10 = 27$, $17 + 11 = 28$, $17 + 12 = 29$, $17 + 13 = 30$, $17 + 14 = 31$
• 18 + T: $18 + 8 = 26$, $18 + 9 = 27$, $18 + 10 = 28$, $18 + 11 = 29$, $18 + 12 = 30$, $18 + 13 = 31$, $18 + 14 = 32$
• 19 + T: $19 + 8 = 27$, $19 + 9 = 28$, $19 + 10 = 29$, $19 + 11 = 30$, $19 + 12 = 31$, $19 + 13 = 32$, $19 + 14 = 33$

Step 3: List Out All Unique Sums

The unique sums you can get from the combinations above are: $\{23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33\}$

Step 4: Count the Number of Different Sums

There are 11 different sums.

Final Answer:

There are 11 different possible sums you can get by choosing one number from $S$ and one number from $T$ and adding them together.

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Would you like more details or have any other questions? Here are five more related questions:

1. How would the number of unique sums change if we added or removed numbers from $S$ or $T$?
2. What would happen if the sets $S$ and $T$ were larger or smaller?
3. Can you find a general formula for the number of unique sums given two sets $S$ and $T$?
4. What would be the effect if the sets $S$ and $T$ had common elements?
5. How does this concept of unique sums apply to problems involving multiple sets or sequences?

Tip: When dealing with sums of elements from two sets, always consider both the range of possible sums and the uniqueness of those sums to avoid counting duplicates.