Question

Assertion (A) : There are $4$ addressed envelopes & $4$ letters for each of them. The probability that no letter mailed in its correct envelope is$\frac{3}{8}$
Reason (R) : The probability that all letters are not mailed correctly is$\frac{23}{24}$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (B)

$n\left(S\right)=4!=24$
Now, number of ways in which all letters go to correct envelop is only one way.
$\therefore$ probability that all letters are correctly placed in right envelop$=\frac{1}{24}$ $\therefore$ Probability that all letters are not placed correctly in the right envelop$=1-\frac{1}{24}=\frac{23}{24}$
$\therefore$ Reason (R) is correct.
Again, the probability that out of $n$ letters & $n$ envelopes none of them enter in the right envelop.
$=\frac{\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}+....+\frac{{\left(-1\right)}^n}{n!}\right)}{n!},n\ge 2$ using $n=4$
P(None of the letter go to the exact envelop) $=\frac{1}{2}-\frac{1}{6}+\frac{1}{24}$
$=\frac{12-4+1}{24}=\frac{9}{24}=\frac{3}{8}$
$\therefore$ Assertion (A) is also correct but Reason (R) is not the proper explanation of the Assertion (A).