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### Corollaries of the Combinations Formula

Total number of combinations of n objects taken r at a time are calculated by the formula . Many other formulas related to combinations can be derived from the above formula. These are termed as 'corollaries'. In total, we are going to discuss five corollaries that can be derived from the above formula.

Corollary 1:
This corollary states that the combinations of n objects taken r at a time are equal to the product of n, (n – 1), (n – 2), .. up-to r factors divided by the factorial of r.
Proof:
 Statement Reason Formula for combinations Canceling (n – r)! from)! from numerator and denominator

### Corollary 2:

This corollary states that there is only one possible combination when all objects are taken at a time. That is, the combinations of objects taken n at a time at a time are 1.
Proof:
 Statement Reason Reason Formula for combinations Substituting r = n because all objects are taken at a time. ()! equals (0)!, which is)!, which is equal to 1. n! divided by n! equals 1.

### Corollary 3:or

This corollary states that the number of combinations of n objects taken r at a time is equal to the number of combinations of n objects taken (n – r) at a time.
Proof:
 Statement Reason Formula for combinations Substituting r = n – r Simplifying the above formula Comparing statements 1 and 3.

### Corollary 4:

This corollary states that if the number of combinations of n objects taken a at a time is equal to the number of combinations of n objects taken b at a time, then either a and b are equal numbers, or the the sum of a and b is equal to the total number of objects, n.
Proof:
 Statement Reason Given From Corollary 3 From statements 1 and 2 From above statement From above statement

### Corollary 5: or

This corollary states that the sum of the number of combinations of n objects taken r at a time and that of n objects taken r – 1 at a time is equal to the number of combinations of n + 1 objects taken r at a time.
Proof:
 Statement Reason Applying the formula of combinations Simplifying the above statement by factoring Adding the fractions Simplifying the above statement