De – arrangement:-
Let there are n different colour boxes & correspondingly n different colour balls.
Now we are to put these balls into boxes (one in each) such that no two colour matches.
= n! – ⁿC₁.(n – 1)! + ⁿC₂.(n – 2)! – ⁿC₃.(n – 3)! + ………(-1)ⁿ ⁿCₙ.(n – n)!
Example:- Let there are 8 addressed envelope & 8 corresponding letters. In how many ways we can put these letters in envelope (one in each) such that
(i). No letter goes to righ envelope
(ii). Exactly 3 letters are placed correctly
(iii). Exactly 3 letters are placed wrongly.
Solution:-
(i).
Answer
(ii). Since exactly 3 letters are placed correctly,
∴ First select 3 letters out of 8 letters. this can be done in ⁸C₃ ways & these letters can be placed correctly only 1 way.
Now remaining 5 letters are de-arranged
Hence total number of ways
= ⁸C₃.5!. Answer
(iii). Since exactly 3-letters are are placed wrongly.
∴ 8 – 3 = 5 letters are placed correctly.
Hence total number of ways
= ⁸C₅.1.3! . Answer