Matching envelopes

Probability Level 1

Five letters ( ( A A to E E ) ) are placed at random in five envelopes marked A A to E E . Where each envelope only contain exactly one letter.

Find the probability that NOT all the letters are put in the matching envelopes.

109 120 \dfrac{109}{120} 119 120 \dfrac{119}{120} 113 120 \dfrac{113}{120} 112 120 \dfrac{112}{120} 115 120 \dfrac{115}{120}

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Munem Shahriar
Jul 1, 2017

The total number of ways of placing 5 5 letters in 5 5 envelopes ⇒ 5 ! = 5 × 4 × 3 × 2 × 1 = 120 \Rightarrow 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 .

All the letters can be placed in the right envelope in only one way. Therefore, the probability that all the letters are placed in the right envelopes is, 1 120 \dfrac{1}{120}

Hence, the probability that all the letters are not placed in the right envelopes is,

⇒ 1 − \Rightarrow 1 - 1 120 \dfrac{1}{120} = 119 120 \dfrac{119}{120}

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