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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
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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



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