You have a collection of 38 distinct songs and you want to create a mix CD that contains exactly 17 different songs. If the order of songs on the CD does not matter (only which songs are included matters), how many different 17-song CDs can be made?

Introduction / Context:
This question is a large scale combinations problem. We are selecting a subset of songs from a larger collection to make a mix CD, and the order of the songs is not important for counting distinct CDs. For such problems, combinations are used, and the numbers involved can become very large. Understanding how to express and compute these combinations is useful in probability, statistics, and combinatorics.

Given Data / Assumptions:

• Total number of distinct songs available: 38.
• We want to include exactly 17 different songs on the CD.
• No song is repeated on the CD.
• Order of songs on the CD is not considered; only the set of songs matters.
• Every choice of 17 distinct songs corresponds to one possible mix CD.

Concept / Approach:
This is a direct combinations problem. We must choose 17 songs from 38 without regard to order. The number of such selections is given by:
nCr = n! / (r! * (n - r)!). Here, n = 38 and r = 17, so the required count is 38C17. This is a large number, but we can still express or compute it using the formula or using symmetry 38C17 = 38C21.

Step-by-Step Solution:
Step 1: Recognize that the answer is 38C17. Step 2: Write 38C17 = 38! / (17! * 21!). Step 3: In practice, this is computed by carefully simplifying the factorial ratio, often using software or a scientific calculator. Step 4: The exact computed value is 28781143380. Step 5: Therefore, there are 28781143380 distinct choices of 17 songs out of 38. Step 6: Each such choice corresponds to a unique unordered mix CD.

Verification / Alternative check:
We can check the reasonableness using symmetry: 38C17 = 38C21. Since the central binomial coefficients grow very large, it is normal that the number is on the order of tens of billions. A calculator that supports nCr will confirm that 38C17 equals 28781143380. Any significantly smaller number would underestimate the true count of combinations.

Why Other Options Are Wrong:
2878114338: This is exactly ten times smaller than the true value and could result from missing a factor of 10 in computation. 29000000000 and 28000000000: These are rough rounded figures that do not match the exact combinatorial calculation of 38C17. Only 28781143380 matches the correct combination value.

Common Pitfalls:
One common mistake is to treat the order of songs as important and use permutations, which gives a massive overcount. Another is to underestimate the size of binomial coefficients and think the answer must be a much smaller number such as thousands or millions. Some students also incorrectly use 38^17, which is not a combination and allows repetition and ordering. Always pay attention to whether order matters and whether repetition is allowed. Here, it is a pure selection problem without repetition, so 38C17 is the correct model.

Final Answer:
The number of different unordered 17 song CDs that can be made from 38 distinct songs is 28781143380.