Find the number of all 6 digit numbers such that All the digits of each number are from the set 1,2,3,4,5 Any digit that appears in the number appeares at least twice.

Step-by-step solution

Case-I: All six digits alike . say 111111, 22222 etc =5 ways Case-II: 2 alike +2 other alike +2 other alike. ^5 C₃ say 1,2,3 and take 11,22,33 aligned & Number of ways = 6! 2!2!2! (in each selection) & Total =10 ( 720 8 )=900 aligned Case-III: 2 alike +4 other alike e.g. 112222 or 221111 etc. Number of ways selecting 2 digits = ( ^5 C ₂ )(2) (in each selection) aligned & Number of ways = 6! 2!4! & Total = ( ^5 C ₂ )(2) ( 6! 2!4! )=(20)(15)=300 aligned Case-IV: 3 alike +3 other alike. Select ^5 C ₂ e.g. 111, 222 etc