Permutations & Combinations
Derangements
MJAT_TS1_P1
Grade 12
Question:
Sudheer has 7 friends. He writes 7 letters, each uniquely meant for one friend, and places them randomly into 7 addressed envelopes, each bearing the specific name of the intended friend.
Out of the seven friends:
1) Two of them are brothers and live at the same address.
2) The other five friends live at five different addresses.
Find the number of ways Sudheer can place the letters into the addressed envelopes such that none of the friends receives the letter meant for them.
(Assume that letters received at the two brothers' address are received by both.)
A) 1280
B) 1368
C) 684
D) 640
Step-by-Step Solution
Key Concept: Treat the two brothers (sharing an address) as a special pair. Count derangements of 7 letters such that neither brother receives their own letter, using inclusion-exclusion: $D_7 - [D_6 + D_6 + D_5]$ where duplicates from the brothers' shared address are accounted for.
Number of valid arrangements $= D_7 - [D_6 + D_6 + D_5]$. Using $D_n = n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}$: $D_7 = 1854$, $D_6 = 265$, $D_5 = 44$. Answer $= 1854 - [265 + 265 + 44] = 1854 - 574 = 1280$.
Correct Answer: A