In an examination, a candidate has to pass in each of the 6 subjects. In how many ways can he fail??

Number of ways in which the candidate can fail in all 6 subjects = 1
Number of ways in which the candidate can fail in 5 subjects = $$^6C_5$$
Number of ways in which the candidate can fail in 4 subjects = $$^6C_4$$
Number of ways in which the candidate can fail in 3 subjects = $$^6C_3$$
Number of ways in which the candidate can fail in 2 subjects = $$^6C_2$$
Number of ways in which the candidate can fail in 1 subject = $$^6C_1$$
So, number of ways in which the candidate can fail = $$^6C_1 + ^6C_2 + ^6C_3 + ^6C_4 + ^6C_5 + ^6C_6 = 2^6 - ^6C_0 = 64 - 1 = 63$$ ways

Every subject has two possibilities either pass or fail.
total number of possibilities for pass and fail= 2*2*2*2*2*2=64
Total number of possibilities for candidate has to pass = 1 possibility i.e. he has to pass all the subjects
No.of possibilities for student fail = 64 - 1=63.