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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.

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