# A clerk was asked to mail three reports to three students . He addresses three envelopes but unfortunately paid no attention to which report card be put in which envelope . what is the probability that exactly one of the students received his or her own card ?

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$\mathrm{We}\mathrm{have}\phantom{\rule{0ex}{0ex}}3\mathrm{reports}\mathrm{and}3\mathrm{envelopes}\phantom{\rule{0ex}{0ex}}\mathrm{Total}\mathrm{arrangement}=3!=6\phantom{\rule{0ex}{0ex}}\mathrm{Now}\phantom{\rule{0ex}{0ex}}\mathrm{we}\mathrm{choose}1\mathrm{report}\mathrm{from}3\mathrm{reports}\mathrm{and}\mathrm{put}\mathrm{it}\mathrm{in}\mathrm{correct}\mathrm{envelope}\mathrm{and}\mathrm{while}\phantom{\rule{0ex}{0ex}}\mathrm{remaining}\mathrm{two}\mathrm{go}\mathrm{in}\mathrm{incorrect}\mathrm{envelope}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Note}\mathrm{that}\mathrm{after}\mathrm{selecting}\mathrm{the}\mathrm{card}\mathrm{which}\mathrm{goes}\mathrm{in}\mathrm{correct}\mathrm{envelope},\mathrm{we}\mathrm{have}\phantom{\rule{0ex}{0ex}}\mathrm{only}\mathrm{have}1\mathrm{way}.\phantom{\rule{0ex}{0ex}}\mathrm{Number}\mathrm{of}\mathrm{ways}{=}^{3}{\mathrm{C}}_{1}\times 1=3\phantom{\rule{0ex}{0ex}}\mathrm{Required}\mathrm{Probability}=\frac{3}{6}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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