Inclusion – Exclusion principle:-
Number of ways = (Total number of ways) – (number of ways when restriction fails atleast once) + (Number of ways when restriction fails at least twice) – (Number of ways when restriction fails at least thrice) + (Number of ways when restriction fails atleast four times) – ……………..
Example:- Let a 9 digit number is to be made which should not contain digits 0, 5 and 7 & must contain digits 1, 2, 4, 8.
Solution:-
Number of ways = 7⁹ – ⁴C₁.6⁹ + ⁴C₂.5⁹ – ⁴C₃.4⁹ + ⁴C₄.3⁹
= 40353607 – 40310784 + 11718750 – 1048576 + 19683
= 10732680 Answer