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Maths Exercices

Binomial coefficient

The number of ways that m things can be 'chosen' from a set of n things is writen as:


and is interpreted as the number of m-element subsets (the m-combinations) of an n-element set, A.

It is called the choose function of n and m, and is defined to be the natural number:

where .

 

Properties of binomial coefficient.


1. Simmetry:

2. Recurrence relation:
3. Newton's binomy.


or similarly:

 


.
Note that:

4. Pascal's triangle

Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle.

A simple construction of the triangle proceeds in the following manner:

  1. On the zeroth row, write only the number 1.
  2. To construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value (If either the number to the right or left is not present, substitute a zero in its place).

Let's see the first six rows of the Pascal's triangle:

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

2

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

3

 

3

 

1

 

 

 

 

 

 

 

 

 

 

 

1

 

4

 

6

 

4

 

1

 

 

 

 

 

 

 

 

 

1

 

5

 

10

 

10

 

5

 

1

 

 

 

 

 

 

 

1

 

6

 

15

 

20

 

15

 

6

 

1

 

 

 


Note that, in Pascal's triangle, the entries on the nth row are given by the binomial coefficients:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Pascal's triangle determines the coefficients which arise in binomial expansions.


Look that the coefficients in this expansion are precisely the numbers on row 5 of Pascal's triangle