https://file.aiquickdraw.com/mathbot/file/17264670722055bfpuj2u.jpg

Let's go through the math problems presented in the image.

1. **Problem 4:**  
   "The product of the first 25 odd numbers can be given as:"

   - 1) \( 49! \)
   - 2) \( \frac{50!}{2^{25}(25!)} \)
   - 3) \( 2^{24}(24!) \)
   - 4) None of these

   The product of the first \(n\) odd numbers is known as a double factorial, which can be written as:
   \[
   \prod_{k=1}^{25} (2k-1) = \frac{50!}{2^{25} \cdot 25!}
   \]
   The correct answer is **Option 2.**

2. **Problem 5:**  
   "Which of the following is the largest?"

   - 1) \( 2500! \times 2500! \)
   - 2) \( 5000! \)
   - 3) \( 3600! \times 1400! \)
   - 4) \( 4000! \times 1000! \)

   The correct answer here will require estimating the relative sizes of factorials. Based on factorial growth rates, we know that **Option 2: \( 5000! \)** is the largest.

3. **Problem 6:**  
   "The product \( 5 \times 10 \times 15 \times 20 \times 25 \times 30 \times 35 \times 40 \times 45 \times 50 \) will end with:"

   - 1) 5 zeroes
   - 2) 6 zeroes
   - 3) 7 zeroes
   - 4) 8 zeroes

   The number of trailing zeroes in a product comes from factors of 10, which is made from multiplying 2s and 5s. Since multiples of 5 give us more 5 factors than 2s, we need to count the number of 5s in the product. For this product, there are 6 factors of 5, so the answer is **Option 2: 6 zeroes.**

4. **Problem 7:**  
   "Which of the following does not completely divide 2001?"

   - 1) \( 72^{41} \)
   - 2) \( 72^{44} \)
   - 3) \( 72^{47} \)
   - 4) \( 72^{50} \)

   To answer this, we need to find the divisibility of powers of 72 for 2001, but based on powers of large numbers, \( 72^{41} \) will not completely divide 2001. Therefore, the answer is **Option 1: \( 72^{41} \).**

Let me know if you'd like more details or have any questions about any of these problems!

### Related questions:
1. How do you calculate factorials for large numbers efficiently?
2. What is the significance of trailing zeroes in factorials?
3. How does the double factorial function differ from regular factorials?
4. Why do powers of numbers like 72 affect divisibility for certain values?
5. How do the growth rates of different factorials compare?

**Tip:** Factorials grow extremely fast, so small differences in their inputs can result in huge output changes!

The product of the first 25 odd numbers, largest factorial comparison, trailing zeroes in product of multiples of 5, and divisibility of powers of 72 by 2001.

Explore complex math problems involving factorial growth, trailing zeroes in products, and divisibility of powers. Learn how to compute factorials and understand their rapid growth, divisibility by large numbers, and how many zeroes result from the multiplication of 5s and 2s.

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