Friday, April 20, 2018

Pascal's triangle

Pascal's triangle

 One of the most interesting triangular array or pattern of numbers (binomial coefficients) is called the Pascal's triangle.
  It is globally named after  Blaise Pascal, a French mathematian.
     
                                      1 
                                 1         1
                             1        2          1
                         1       3        3         1
                      1      4       6           4     1
                  1      5      10      10        5      1
   
  There is a rule for the element/entry in m- th row and n- th column of Pascal's triangle.
  The element/entry is: m!/[n! ×(m-n)! ].
   There are many interesting facts about the Pascal's triangle:
    (1): The horizontal sums of each row is a power of 2.
         
Pascal triangle
    (2): each horizontal line/row of Pascal's triangle is a power of 11.
         

Pascal triangle
Note: but for 6 th line the digits overlaps.
i.e. 15101051 = 1(5+1)(0+1)051=161051.
(3): The sum of diagonal elements of  Pascal's triangle represents the Fibonacci sequence.
(4): The interesting fact is Pascal's triangle gives the combinations of heads and tails in a toss of with a  coin. Not only that, it also gives us the probability of getting any no. of heads exactly.
As for example, if we toss a coin three times; the combinations of heads and tails are: HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. Which is in the pattern: 1, 3,3,1.
Also we can obtain the probability of getting exactly two heads as follows:
There are total (1+3+3+1=8) outcomes or event points.(also, 2³=8). And no. of event points with exactly two heads is 3.
So, the probability of getting exactly 3 heads is: 3/8, which is also obvious result by the theory of Probability.
(5): If we observe the diagonals of Pascal's triangle, we can see that:
The first diagonal is a sequence of unity(1), The second is a sequence of Natural numbers(1,2,3,..), The third diagonal is a sequence of triangular numbers (1,3,6,10,...) and four is a sequence of tetrahedral numbers.
(6):The Pascal's triangle is symmetrical on both sides (left and right) like a mirror image.
Pascal triangle


  
 If you find out any incorrect information or know anything more about this , please write it in the comment section!
    


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