​​i how many permutations of 8 different things taken 5 at a time will 2 particular thing  will never come together

Dear student

Total permutation of 8 different things taken 5 at a time 

$={}_5{}^{8}P$

Now if 2 particular things do not come together

let us assume that 2 things are coming together and take them combined as 1

so when they come together, we will have out of 7 different things, we have to take 4 at a time (as one thing is the combination of two things mentioned above)

Total number of permutations so that they never come together

= Total permutation of 8 different things taken 5 at a time - ​Total permutation of 7 different things taken 4 at a time -


$$\begin{gathered} ={}_5{}^{8}P-{}_4{}^{7}P \\ =\frac{8!}{5!*3!}-\frac{7!}{4!*3!} \\ =56-35 \\ =21 \end{gathered}$$

Regards