Given: There are 5 subjects and for a student to pass in an examination the students has to pass in each of the 5 subjects. Here, we have to find in how many ways a student can fail in the examination. In order to fail in the examination the student can fail in 1 or 2 or 3 or 4 or 5 subjects out of the 5 subjects in each case. Case 1: Student fails in any 1 subject out of the 5 subjects. No. of ways in which student can fail in any 1 subject out of the 5 subjects = 5C1 Case 2: Student fails in any 2 subjects out of the 5 subjects. No. of ways in which student can fail in any 2 subjects out of the 5 subjects =5C2 Case 3: Student fails in any 3 subjects out of the 5 subjects. No. of ways in which student can fail in any 3 subjects out of the 5 subjects =5C3 Case 4: Student fails in any 4 subjects out of the 5 subjects. No. of ways in which student can fail in any 4 subjects out of the 5 subjects =5C4 Case 5: Student fails in all 5 subjects. No. of ways in which student can fail all 5 subjects. =5C5 ∴ Total number of ways in which a student can fail in an examination = ‌5C1+‌5C2+‌5C3+‌5C4+‌5C5=31