Combinatorics

In the examination, there are 9 papers, and the candidate must pass in more papers than the number of papers he fails in.

- There are two possibilities for each paper: either he passes or he fails.

- The total number of possible outcomes (ways of passing or failing) for all 9 papers is .

To be unsuccessful, the candidate must fail in at least 5 papers (because he needs to pass in more papers than he fails). Thus, we need to find the number of outcomes in which he fails in 5 or more papers.

- If he fails in 5 papers, there are ways to choose which 5 papers he fails.

- Similarly, for failing in 6, 7, 8, or 9 papers, the number of ways are .

The total number of ways he can be unsuccessful is the sum of all these outcomes: This sum is equal to , which is the total number of ways he can be unsuccessful.

Thus, the correct answer is .