Question

There are five letters and five addressed envelopes corresponding to them. The number of ways the letters can be placed (one in each envelope) such that no letter is in its corresponding envelope is

• A. 44.00
• B. 044.0
• C. 44
• D. 044
• E. 44.0

Answer 1

Answer: (A), (B), (C), (D), (E)

There is only $1$ way that all five letters are in their corresponding envelope.
If no letters are in their corresponding envelope then we have to find out how all the letters can be deranged.
$D_n=n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots +(-1{)}^n\frac{1}{n!})$ where $n>2$
Here $n=5,$
$\Rightarrow D_5=5!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})\Rightarrow D_5=44$