Let's determine the mode of the number of years served by U.S. Supreme Court Justices. The mode is the number that appears most frequently in a data set.
To find the mode, we need to examine the list of years served:
[tex]\[
\begin{array}{llllllllll}
0 & 4 & 6 & 8 & 13 & 15 & 18 & 22 & 26 & 31 \\
1 & 4 & 6 & 9 & 13 & 15 & 19 & 23 & 26 & 32 \\
1 & 4 & 6 & 9 & 14 & 16 & 19 & 23 & 27 & 33 \\
2 & 5 & 7 & 9 & 14 & 16 & 20 & 23 & 28 & 33 \\
2 & 5 & 7 & 10 & 14 & 16 & 20 & 23 & 28 & 33 \\
2 & 5 & 7 & 10 & 14 & 16 & 20 & 23 & 28 & 34 \\
3 & 5 & 7 & 10 & 15 & 17 & 20 & 24 & 29 & 34 \\
3 & 5 & 8 & 11 & 15 & 17 & 20 & 24 & 30 & 34 \\
3 & 5 & 8 & 13 & 15 & 18 & 21 & 26 & 30 & 36 \\
4 & 5 & 8 & 13 & 15 & 18 & 21 & 26 & 31 &
\end{array}
\][/tex]
By inspecting the list, we count the frequency of each number of years:
- 0: 1 time
- 1: 2 times
- 2: 3 times
- 3: 3 times
- 4: 6 times
- 5: 10 times
- 6: 4 times
- 7: 5 times
- 8: 6 times
- 9: 4 times
- 10: 4 times
- 11: 1 time
- 13: 6 times
- 14: 5 times
- 15: 7 times
- 16: 5 times
- 17: 2 times
- 18: 4 times
- 19: 3 times
- 20: 5 times
- 21: 2 times
- 22: 1 time
- 23: 6 times
- 24: 2 times
- 26: 6 times
- 27: 1 time
- 28: 4 times
- 29: 1 time
- 30: 3 times
- 31: 3 times
- 32: 1 time
- 33: 4 times
- 34: 3 times
- 36: 1 time
From the counts above, we can see that the number 5 appears 10 times, which is more frequent than any other number.
Therefore, the mode of the number of years served by U.S. Supreme Court Justices is:
[tex]\[
\boxed{5}
\][/tex]
