Mathematics - Permutation & Combination Question with Solution | TestHub

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 $2367 \& 154$ 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.