Let's break down the problem:

You have two sets:

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

Each set has 5 and 7 elements, respectively. You are asked to find how many different sums can be made by choosing one element from each set and adding them together.

Step-by-Step Solution:

1. Find the range of possible sums:

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

2. List all possible sums:

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

3. Determine the unique sums:

   • The unique sums are: $\{23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33\}$.

4. Count the number of unique sums:

   • There are 11 different sums.

Conclusion:

The number of different sums you can get by adding one element from each set is 11.

Would you like a detailed explanation or have any further questions?

Related Questions:

1. What happens if we change the range of elements in set $S$?
2. How many different sums can be formed if both sets had the same number of elements?
3. What if we include negative numbers in either set?
4. How does the number of elements in each set affect the number of unique sums?
5. Can we find a general formula to calculate the number of different sums?

Tip:

When dealing with sums from two sets, always consider the smallest and largest possible sums to set the boundaries for possible outcomes.