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).