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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.
Solution
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
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