To determine which set of numbers is ordered from the least to greatest, let's evaluate the four numbers given:
1. Convert each number to its decimal form:
- \(1 \frac{3}{4} = 1 + \frac{3}{4} = 1 + 0.75 = 1.75\)
- \(\sqrt{\pi} \approx 1.7724538509055159\)
- \(1.71\) (already in decimal form)
- \(\frac{16}{9} \approx 1.7777777777777777\)
2. Arrange these decimal values from least to greatest:
- \(1.71\)
- \(1.75\) (which is \(1 \frac{3}{4}\))
- \(1.7724538509055159\) (which is \(\sqrt{\pi}\))
- \(1.7777777777777777\) (which is \(\frac{16}{9}\))
3. Compare the given answers to see which matches the order \(1.71, 1 \frac{3}{4}, \sqrt{\pi}, \frac{16}{9}\):
- First option: \(1 \frac{3}{4}, \sqrt{\pi}, 1.71, \frac{16}{9}\) - does not match the correct order.
- Second option: \(\sqrt{\pi}, 1 \frac{3}{4}, \frac{16}{9}, 1.71\) - does not match the correct order.
- Third option: \(1.71, 1 \frac{3}{4}, \sqrt{\pi}, \frac{16}{9}\) - matches the correct order.
- Fourth option: \(\frac{16}{9}, 1.71, \sqrt{\pi}, 1 \frac{3}{4}\) - does not match the correct order.
Thus, the correct answer choice has the set of numbers ordered correctly from least to greatest is:
[tex]\[ \boxed{1.71, 1 \frac{3}{4}, \sqrt{\pi}, \frac{16}{9}} \][/tex]
