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!
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