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Pascal’s Triangle.

n is the row number and  sum is the sum of the numbers in the row.

n   sum
0    1                            1
1    2                         1     1                     
2    4                      1     2      1
3    8                   1    3     3      1
4   16                1   4      6     4      1
5   32             1    5    10   10     5     1
6   64          1    6   15    20    15    6     1   
7  128      1   7    21   35    35   21    7      1

Sum of numbers in the row  = Sum of the coefficients of binomial expansion = 2^n

Sum of coefficients in a row :   2^n. The numbers in the row are   coefficients of terms in  a and b  in the expansion of  (a + b)^n.

The number of   numbers in the row  are  equal to   n + 1.  We start with n = 0.

     The binomial coefficients are used a lot in the permutations and combinations and probability calculations. Sum of the squares of the numbers in a row  “n”  equals the middle element  of the  row number  “2n”. The Pascal’s triangle and the numbers have some uses in defining complicated mathematical functions, some cellular automation related functions.

     There are operations defined on Pascal’s triangle using matrices.
  It seems the famous Indian mathematician  Pingala knew the Pascal’s triangle in 2nd Century BC.   Mahavira  , Jain mathematician also gave a formula for enumerating them in 8th century.   Persian mathematicians too did work on during 10th century AD.   The Chinese were also aware of the Pascal’s triangle in 11th century (Jia Xuan).

  Pascal’s triangle was published in 1665 with a lot of facts and with solved problems in probability theory.    Pascal was a French scientist, mathematician.

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