Pascal's Triangle

Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it in the previous row.

Various patterns can be observed in Pascal's Triangle, some of which are

Structure

• The triangle starts with a 1 at the top.
• Each row begins and ends with 1.
• Any other number in the triangle is the sum of the two numbers immediately above it in the previous row.

Diagonals in Pascal’s Triangle

Each rightward diagonal of Pascal’s Triangle, when considered as a sequence represents the different numbers such as

• First rightward diagonal represents a sequence of number 1.
• Second rightward diagonal represents triangular numbers.
• Third rightward diagonal represents the tetrahedral numbers.
• Fourth rightward diagonal represents the Penelope numbers and so on.

Pascal’s Triangle Binomial Expansion

We can easily find the coefficient of the binomial expansion using Pascal's Triangle. The elements in the (n + 1)th row of the Pascal triangle represent the coefficient of the expanded expression of the polynomial (x + y)ⁿ.

We know that the expansion of (x + y)ⁿ is,
(x + y)ⁿ = a₀xⁿ + a₁x^n-1y + a₂x^n-2y² + … + a_n-1xy^n-1 + a_nyⁿ

Here, a₀, a₁, a₂, a₃, ...., a_n are the term in the (n+1)th row of Pascal's Triangle

Horizontal Sum of Rows

On close observing Pascal’s Triangle we can conclude that the sum of any row in Pascal’s triangle is equal to a power of 2. The formula for the same is, For any (n + 1)^th row in Pascal’s Triangle the sum of all the elements is, 2ⁿ

Applying this Formula in the first 4 rows of Pascal’s triangle we get,

1 = 1 = 2⁰
1 + 1 = 2 = 2¹
1 + 2 + 1 = 4 = 2²
1 + 3 + 3 + 1 = 8 = 2³

Exponents of 11

Each row of Pascal's Triangle corresponds to a power of 11:

Row n (starting from n = 0) represents 11 ⁿ.

Examples:

1. Row 0: 1 → 11⁰ = 1
2. Row 1: 1, 1 → 11¹ = 11
3. Row 2: 1, 2, 1 → 11² = 121
4. Row 3: 1, 3, 3, 1 → 11³ = 1331

Note: For higher powers of 11, the digits in the triangle rows exceed single digits, causing a carry-over effect. In such cases, the numbers in Pascal's Triangle no longer directly represent 11ⁿ. Instead, the pattern can still be reconstructed by summing the digits appropriately.

Fibonacci Sequence in Pascal’s Triangle

We can easily obtain the Fibonacci sequence by simply adding the numbers in the diagonals of Pascal's triangle. This pattern is shown in the image added below,

Pascal’s Triangle Formula

Pascal Triangle Formula is the formula that is used to find the number to be filled in the m^th column and the n^th row. As we know the terms in Pascal's triangle are the summation of the terms in the above row. So we require the elements in the (n-1)^th row, and (m-1)^th and nth columns to get the required number in the m^th column and the nth row.

The elements of the nth row of Pascal's triangle are given, ⁿC₀, ⁿC₁, ⁿC₂, ..., ⁿC_n. 

The formula for finding any number in Pascal's triangle is:

ⁿC_m = ^n-1C_m-1 + ^n-1C_m

Where,

• ⁿC_m represents the (m + 1)th element in the nth row., and
• n is a non-negative integer [0 ≤ m ≤ n]

Pascal’s Triangle in Coin Toss Probability

We use the Pascal triangle to find the various cases of the possible outcomes in probability conditions. This can be understood by the following example, tossing a coin one time we get two outcomes i.e. H and T this is represented by the element in the first row of Pascal's Triangle.

Similarly tossing a coin two times we get three outcomes i.e. {H, H}, {H, T}, {T, H}, and {T, T} this condition is represented by the element in the second row of Pascal's Triangle.

Thus, we can easily tell the possible number of outcomes in tossing a coin experiment by simply observing the respective elements in the Pascal Triangle.

The table below tells us about the cases if a coin is tossed one time, two times, three times, and four times, and its accordance with Pascal's Triangle

Properties

Various Properties of Pascal's Triangle are,

• Every number in the Pascal triangle is the sum of the number above it.
• The starting and the end numbers in Pascal's triangle are always 1.
• The first diagonal in Pascal's Triangle represents the natural number or counting numbers.
• The sum of elements in each row of Pascal's triangle is given using a power of 2.
• Elements in each row are the digits of the power of 11.
• The Pascal triangle is a symmetric triangle.
• The elements in any row of Pascal's triangle can be used to represent the coefficients of Binomial Expansion.
• Along the diagonal of Pascal's Triangle, we observe the Fibonacci numbers

Practice Questions on Pascal's Triangles