The number of permutations, of the digits 1, 2, 3, , 7 without...

Given,

The number $1, 2, 3, . . . . , 7$

Now total numbers of $7$ digit number will be $7 !$ without repetition,

Now total numbers which contain string $154$ will be $5 !$

Total numbers which contain string $2367$ will be $4 !$

And number which contain both string $2367 \text{ and } 154$ will be $2$

So, total number which does not contain string $154$ or $2367$ will be $7 ! - \left(5 ! + 4 ! - 2\right)$

$= 5040 - \left(120 + 24 - 2\right)$

$= 5040 - 142 = 4898$

Hence this is the required answer.