The combinations formula is

`C(n, r) = {n!}/{(n - r)! * r!}`in which

- n is the total number of objects to choose from
- r is the number of objects in each selection
- ! means factorial of the number preceding it
- C(n, r) is the total number of combinations of 'n' objects taken 'r' at a time

The above formula is derived
as follows:

Let there be 'n' objects from which you have to form selections or groups of 'r' objects at a time. Let the total number of combinations, or selections of these 'n' objects taken 'r' at a time be `C(n, r)`. Now each of these `C(n, r)` combinations contains r objects, and hence gives rise to r! different permutations. Thus the total number of permutations of all of these n objects, taken r at a time is equal to `C(n, r) * r`.

Now, we know that the
formula for permutations of n objects
taken r at
a time is `{n!}/{(n - r)! * r!}`.
Hence `C(n, r) * r` is
equal to `{n!}/{(n - r)! * r!}`. Thus we obtain
the equation, `C(n, r) * r = {n!}/{(n - r)! * r!}`.
Solving this equation for C(n, r),
we obtain:

`C(n, r) = {n!}/{(n - r)! * r!}`which is the formula for combinations of n objects taken r at a time.

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