# Some Important Derivations

While deriving an expression for*P*

^{n}*,*

_{r}_{ }we imposed two constraints, viz. distinct things and repetition being not allowed over it and learned how to find the number of permutations. Let us now see what will happen if we do not impose these two restrictions on

*P*

^{n}*.*

_{r}*Number of arrangements of n things of which p are of one type, q are of a second type and the rest are distinct*

**When all the things are not distinct, then we cannot use the general formula for**

*P*

^{n}*for any value of*

_{r}*r*. If we want to find out

*P*

^{n}*for a specific value of*

_{r}*r*in that given problem, we will be required to use it on the basis of the given situation.

The number of ways in which

*n*things may be arranged taking all of them at a time, when*p*of the things are exactly alike of one kind,*q*of them exactly alike of another kind,*r*of them exactly alike of a third kind, and the rest all is distinct isNumber of permutations of n distinct things where each one of them can be used for any number of times (i.e., repetition allowed)

**Derivation for this is based upon common sense. If I have 5 friends and 3 servants and I have to send the invitation letters to all my friends through any of my servants, I obviously have 3 options for the invitation card to be sent to friend1, the same 3 options for the invitation card to be sent to friend2, and similarly 3 options for the invitation card to be sent to each of the friends. So the total number of ways of sending the invitation letters = 3**^{5}And it will not be 5^{3}, as friends are not going to the servants to get the letter.In general, the number of perambulations of

*n*things, taking*r*at a time when each of the thing may be repeated once, twice,â€¦up to*r*times in any arrangement is*n*.^{r}