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In an examination a candidate has to pas...

In an examination a candidate has to pass each of the 5 subiects.The number of ways can be fail is

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The candidate can fail by failing in 1 or 2 or 3 or 4 or 5 subjects out of 5 in each case.
`:.` the total number of ways in which he can fail
`=.^(5)C_(1)+.^(5)C_(2)+.^(5)C_(3)+.^(5)C_(4)+.^(5)C_(5)`
`=.^(5)C_(1)+.^(5)C_(2)+.^(5)C_(2)+.^(5)C_(1)+.^(5)C_(5) [:' .^(n)C_(r)=.^(n)C_(n-r)]`
`=(5+10+10+5+1)=31`.
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