Introduction

In order to determine the number of ways a candidate can fail an examination, we need to consider the various possibilities and combinations in which the candidate can fail each of the four papers.

Analysis

To calculate the number of ways a candidate can fail, we need to consider the following scenarios:

Scenario 1: Failing in one paper
- The candidate can fail in any one of the four papers while passing the remaining three papers.
- The number of ways the candidate can fail in one paper is given by the combination formula: C(4,1) = 4

Scenario 2: Failing in two papers
- The candidate can fail in any two out of the four papers while passing the remaining two papers.
- The number of ways the candidate can fail in two papers is given by the combination formula: C(4,2) = 6

Scenario 3: Failing in three papers
- The candidate can fail in any three out of the four papers while passing only one paper.
- The number of ways the candidate can fail in three papers is given by the combination formula: C(4,3) = 4

Scenario 4: Failing in all four papers
- The candidate can fail in all four papers.
- The number of ways the candidate can fail in four papers is given by the combination formula: C(4,4) = 1

Total number of ways to fail

To find the total number of ways the candidate can fail, we need to sum up the number of ways from each scenario:

Total number of ways to fail = Number of ways to fail in one paper + Number of ways to fail in two papers + Number of ways to fail in three papers + Number of ways to fail in four papers

Total number of ways to fail = 4 + 6 + 4 + 1 = 15

Therefore, there are 15 ways in which a candidate can fail the examination.

Conclusion

In conclusion, the number of ways a candidate can fail an examination, while passing in each of the four papers, is 15. This calculation is based on the different scenarios in which the candidate can fail in one, two, three, or all four papers.