3 letters are placed randomly into 3 envelope | vedclass

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(A) Let the letters be $L_1, L_2, L_3$ and their corresponding envelopes be $E_1, E_2, E_3$.The total number of ways to place $3$ letters in $3$ envel

Vedclass · Accepted Answer

$1/2$

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(A) Let the letters be $L_1, L_2, L_3$ and their corresponding envelopes be $E_1, E_2, E_3$.The total number of ways to place $3$ letters in $3$ envelopes is $3! = 6$.The possible arrangements are: $(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)$.We want exactly one letter in the correct envelope.If $L_1$ is in $E_1$,then $L_2$ and $L_3$ must be swapped ($L_2$ in $E_3$,$L_3$ in $E_2$). This is $1$ way.If $L_2$ is in $E_2$,then $L_1$ and $L_3$ must be swapped. This is $1$ way.If $L_3$ is in $E_3$,then $L_1$ and $L_2$ must be swapped. This is $1$ way.Total favorable ways $= 1 + 1 + 1 = 3$.The probability is $\frac{	ext{Favorable ways}}{	ext{Total ways}} = \frac{3}{6} = \frac{1}{2}$.

$3$ letters are placed randomly into $3$ envelopes. What is the probability that exactly one letter is placed in the correct envelope?

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