Permutation of natural numbers Find the number of permutation of {1,2,3,4,5,6} such that the patterns 13 and 246 do not appear. Show the steps .

It is easier to first find the number of permutations in which the pattern $13$ or $246$ does appear.
For the pattern $13$, tie $1$ and $3$ together to make a supersymbol. Together with $2,4,5,6$, and the supersymbol. we have $5$ "symbols" that can be arranged in $5!$ ways. 
Similarly, there are $4!$ ways in which the pattern $246$ appears.
There are $3!$ ways in which both patterns appear. These are counted twice in the sum $5!+4!$. 
Thus there are $5!+4!-3!$ ways thast one or both of our patterns can appear. It follows that there are $6!-(5!+4!-3!)$ ways to arrange our symbols so that neither pattern appears. 
