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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
A person writes letters to 6 friends and addresses the corresponding envelopes.
Let $x$ be the number of ways so that **at least 2 letters** are in wrong envelopes and
$y$ be the number of ways so that **all letters** are in wrong envelopes.
Then find $x - y$.
Solution
Total ways to arrange 6 letters = $6! = 720$. Let $D_6$ = number of derangements (no letter in correct envelope): $D_6 = 6! \left(1 - \dfrac{1}{1!} + \dfrac{1}{2!} - \dfrac{1}{3!} + \dfrac{1}{4!} - \dfrac{1}{5!} + \dfrac{1}{6!}\right) = 265.$ Now, $x =$ number of ways with at least 2 letters wrong $= 6! - [\text{exactly 0 or 1 correct letters}]$ For exactly 0 correct = $D_6 = 265$. For exactly 1 correct: choose 1 correct letter $(6C1)$ × derange remaining 5 $(D_5)$. $D_5 = 44$, so ways = $6 \times 44 = 264.$ Hence, $x = 720 - (1 + 264) = 455.$ $\therefore x - y = 455 - 265 = 190.$Online Test Series, Information About Examination,
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