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.

         

    (2): each horizontal line/row of Pascal's triangle is a power of 11.

         

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.

  

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