Question

A student passes an examination if he secures a minimum in each of the 7 subjects. Find the number of ways a student can fail.

Answer 1

A student fail, if he does not secure the minimum marks in either 1, 2, 3, 4, 5, 6 or 7 subjects.

∴ the total number of ways in which a student can fail

= ⁷C₁ + ⁷C₂ + ⁷C₃ + ⁷C₄ + ⁷C₅ + ⁷C₆ + ⁷C₇

= `7 + (7 xx 6)/(2 xx 1) + (7 xx 6 xx 5)/(3 xx 2 xx 1) + (7 xx 6 xx 5)/(3 xx 2 xx 1) + (7 xx 6)/(2 xx 1) + 7 + 1`

= 7 + 21 + 35 + 35 + 21 + 7 + 1

= 127

Alternative method:

For each subject, a student has two possible outcomes-to secure minimum marks or he does not.

∴ total number of outcomes for 7 subjects

= 2 × 2 × 2 × ... 7 times

= 2⁷

= 128

Of these there is only one outcome in which he secures the minimum marks in all subjects to pass.

Hence, the number of ways a student can fail

= 128 – 1

= 127