In the permutations of $n$ things $r$, taken together, the number of permutations in which $m$ particular things occur together is ${ }^{n-m} P_{r-m} \times{ }^r P_m$.

In a steamer there are stalls for 12 animals and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in $3^{12}$ ways.

If some or all of $n$ objects are taken at a time, then the number of combinations is $2^n-1$.

There will be only 24 selections containing atleast one red ball out of a bag containing 4 red and 5 black balls. It is being given that the balls of the same colour are identical.

Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on other side of the table. The number of ways in which the seating arrangements can be made is $\frac{11!}{5!6!}(9!)(9!)$.