Nothing is in its place!!!

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Consider a set of a large number of pairs of envelopes and letters. The $n^{th}$ envelope and letter is denoted by $E_{n}$ and $L_{n}$ respectively and both of them have the number $n$ printed on them.
All the letters have been taken out of the envelopes and now are arranged randomly.

Find the probability that the letter $L_{n}$ does not go in envelope $E_{n}$ $\forall n \in \mathbb{N}$

Details and Assumptions:
By a large number, I mean the total number of pairs approach infinity.

The answer is 0.367879.

1 solution

Abhi Singh
Mar 2, 2014

If ‘n’ particular items need to be arranged or organized in such a manner that none of them acquires its usual or correct place is calculated using the formula: n! ( 1 - (1/1!) + (1/2!) - (1/3!) + ....((-1)^n)*(1/n!)

now the total arrangements of n letters in n alphabets is given by n! so probability =n! [ ( 1 - (1/1!) + (1/2!) - (1/3!) + ....((-1)^n) (1/n!) ] / ( n! ) =( 1 - (1/1!) + (1/2!) - (1/3!) + ....((-1)^n) (1/n!)

now expansion of e^x = 1 + x/( 1! ) + (x^2)/( 2! ) + x^3/( 3! )+.... x^n/( n! ) putting x= -1 we get the required probability so probability = e^( -1 ) = 0.3678

Nothing is in its place!!!

The answer is 0.367879.

1 solution

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