Question

There are $5$ address envelopes. Letters are written to corresponding person but the letter are placed at random in envelopes, the probability for none of the letter enter into the correct envelope is?

• A. $\frac{1}{5!}$
• B. $\frac{11}{120}$
• C. $\frac{11}{30}$
• D. None of these

Answer 1

The correct option is C $\frac{11}{30}$

Total possible arrangement=5!=120

Formula for derangement=Dearr(n) = n!(1/0! - 1/1! + 1/2! - 1/3!.... 1/n!)

Dearr(5) = 5! (1/0! - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!)
=> Dearr(5) = 120 (1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
=> Dearr(5) = 60 - 20 + 5 - 1 = 44

Probability =44/120=11/30