Derivation of the Combinations Formula

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.