In how many permutations of the digits 0 1 2 9 is a 4 immed

In how many permutations of the digits 0, 1, 2, ..., 9 is a 4 immediately followed by 2, or 0 immediately followed by 4, or 6 immediately followed by 0? For example, none of the pairs 42, 04, 60 appears in the permutation 7295413806.

Solution

Let P₄₂ be the permutation in which the pair 42 appears

consider permutations 42, 0,,1,3,5,6,7,8,9 there are 9 digits

permuttion of these 9digits = 9 ! -------(1)

consider permutations 04, 1,2,3,5,6,7,8,9, there are 9 digits

permutation of these digits = 9 ! ( which include 042 coming together )

permutation of theses digits where 042 together is removed = 9! - 8! - 7! (6042 together)---(2)

similarly 60 togethrer = 9! - 8! - ! ( 6042 together )------(3)

from 1,2,3 total no of permutations = 9! +9!+9! -8!-8!-7!-7!= 3 (9!) - 2(8!) -2(7!)