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### 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.