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Pascal’s Triangle

 

Pascal’s triangle is a concept of Mathematics. Named after the French mathematician Blaise Pascal, it was first made in use by the Chinese mathematician Jia Xian and later studied and popularized by many other mathematicians.

Pascal’s triangle, in general, is the arrangement of numbers in a pattern that resembles a triangle. The numbers are basically binomial coefficients obtained through different probabilities. These binomial coefficients are then placed one below another to form a triangular array. The first element of Pascal’s triangle is always 1 which is placed in the first row on the top of the triangular pattern followed by other numbers place below each other in different rows.

Constructing a Pascal’s triangle

The basic rule to construct Pascal’s triangle is to maintain its triangular shape. It has 1 as its first element. 1 is placed on the top of the triangle in the first line. However, 1 is also used in repetitive patterns while constructing the triangle. For that, a pattern of 1s is used in the corners of each row of Pascal’s triangle. Every row begins with element 1 and also ends with 1. This also gives the triangle’s sides an arrangement of 1s as we proceed down a row from the beginning to the end. The base of the triangle also consists of 1s with each 1 at the starting of the row and at the end of the row. Therefore, the first row of Pascal’s triangle has only one element i.e. 1. For the second row, a number of elements become two. This is calculated by adding the number on the left side of the first row. It gives the first number of the second row. For the second number of the second row, add the number on the right side of the first row. Hence, for the second row, we get the elements as 1 and 1. For example, consider the below given Pascal’s triangle

1

1  1

We notice that there is no number on the left of the first row. So, 0 + 1 gives 1 which is the first element of the second row. Similarly, there is no element on the right of the first row. This means 0 + 1 gives 1 which is the second element of the second row. Now, proceed in the same manner for the third row and so on.

1  row1

1  1  row2

1  2  1  row3

1  3  3  1  row4

1  4  6  4  1  row 5

1  5  10  10  5  1  row 6

For row 3, 0 + 1 = 1 and 1 + 1 = 2

For row 4, 0 +1 =1 and 1 + 2 = 3

For row 5, 0 + 1 =1, 1 + 3 = 4 and 3 + 3 = 6

For row 6, 0 + 1 = 1, 1 + 4 =5 and 5 + 5 = 10

Pascal’s triangle in probability

To construct Pascal’s triangle in probability, consider the case of tossing a coin for ‘n’ number of times.

For n = 1

{H}

{T}

For n = 2

{HH}

{HT, TH}

{TT}

For n = 3

{HHH}

{HTH, HHT, THH}

{THT, TTH, HTT}

{TTT}

and so on.

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