Suppose you want to find the solution to a traveling salesman problem with 19 cities. a. How many Hamiltonian circuits will need to be tested? b. With a computer that can test 1 million circuits per second, how long will the job take? c. Put the answer to part b in perspective by comparing it to some well-known length of time. a. About \( 3.2 \times 10^{15} \) circuits will need to be tested. (Use scientific notation. Round to the nearest tenth as needed. Use the multiplication symbol in the math palette as needed.) b. Using the answer from part a, the job should take (Type an integer or a decimal.)

**Step 1. Calculate the total number of Hamiltonian circuits** For a symmetric traveling salesman problem with \( n \) cities, the number of Hamiltonian circuits is given by \[ \frac{(n-1)!}{2}. \] For \( n = 19 \), we have \[ \frac{18!}{2}. \] We are given that this is approximately \[ 3.2 \times 10^{15} \text{ circuits}. \] **Step 2. Calculate the time required to test all circuits** If the computer tests \( 1 \times 10^6 \) (one million) circuits per second, the time \( T \) in seconds needed is \[ T = \frac{3.2 \times 10^{15}}{1 \times 10^6} = 3.2 \times 10^9 \text{ seconds}. \] **Step 3. Convert the time into years** There are approximately \( 3.15 \times 10^7 \) seconds in a year (since \( 1 \) year \( \approx 365 \) days). Thus, the time in years is \[ \frac{3.2 \times 10^9 \text{ seconds}}{3.15 \times 10^7 \text{ seconds/year}} \approx 101.6 \text{ years}. \] Rounded to the nearest tenth, the job should take about \[ 101.6 \text{ years}. \] **Step 4. Compare with a well-known length of time** A typical human lifetime is about 80 years, so 101.6 years is longer than a human lifetime. Additionally, it is roughly equivalent to slightly more than a century. This puts the computational effort into perspective as it would take over 100 years of continuous computation. **Final Answers:** a. \(3.2 \times 10^{15}\) circuits b. \(101.6\) years c. The job would take more than a century to complete, which is far beyond the scale of a typical human lifetime.