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),
.. upto r factors divided by the factorial of r.Corollary 1:
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  
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