# Lesson Remarkable identities for Binomial Coefficients

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## Remarkable identities for Binomial Coefficients

### Problem 1

Prove this identity for  Binomial Coefficients

+ + + . . . + + . . . + =

for any positive integer number  n.

Solution
The proof is very straightforward.  Simply substitute   =   into the  Binomial Expansion

=   =

(see the lesson  Binomial Theorem  under the current topic in this site).  You will get the required identity.

This is how it looks for the low order binomial coefficients:

n = 1,   = 1,   = 1,   + = 1 + 1 = 2;

n = 2,   = 1,   = 2,   = 1,   + + = 1 + 2 + 1 = 4;

n = 3,   = 1,   = 3,   = 3,   = 1,   + + + = 1 + 3 + 3 + 1 = 8.

### Problem 2

Prove this identity for  Binomial Coefficients

- + + . . . + + . . . + =

(alternate sum) for any positive integer number  n.

Solution
Again, the proof is very straightforward.  Simply substitute   =   into the  Binomial Expansion

=   =

(see the lesson  Binomial Theorem  under the current topic in this site).  You will get the required identity.

This is how it looks for the low order binomial coefficients:

n = 1,   = 1,   = 1,   - = 1 - 1 = 0;

n = 2,   = 1,   = 2,   = 1,   - + = 1 - 2 + 1 = 0;

n = 3,   = 1,   = 3,   = 3,   = 1,   - + - = 1 - 3 + 3 - 1 = 0.

### Problem 3

Prove this identity for  Binomial Coefficients

+ + + . . . + + . . . + =

for any positive integer number  n.

Solution
Similar to the Problem 1 and Problem 2 above, the proof is very straightforward.  Simply substitute   =   into the  Binomial Expansion

=   =

(see the lesson  Binomial Theorem  under the current topic in this site).  You will get the required identity.

By doing in a similar way,  you can prove many other identities for Binomial Coefficients,  like these

+ + + . . . + + . . . + = ,

+ + + . . . + + . . . + = ,

- + + . . . + + . . . + = .

For the close subject on the  Pascal's triangle  see the lesson  The Pascal's triangle  under the current topic in this site.

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