Why is Pascal’s Triangle important for JEE?

Pascal’s Triangle simplifies binomial expansions, combinatorial calculations, and helps in solving probability, series, and pattern-based questions frequently asked in JEE exams.

Can Pascal’s Triangle be used for negative or fractional exponents?

The classic triangle is for non-negative integers, but the generalized binomial theorem handles non-integers using infinite series.

How is Pascal’s Triangle applied in probability?

Binomial coefficients from the triangle are used to compute probabilities in binomial and related distributions.

What is the hockey stick identity?

It’s a property for summing consecutive combinations diagonally, giving a single combination value further down the triangle.

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Pascal’s Triangle

1.0What is Pascal's Triangle?

Pascal's Triangle is a triangle of numbers where each number is the sum of the two numbers above it. Each row shows the coefficients of the binomial expansion for higher powers of a binomial expression.

The nth row (starting from n = 0) contains the binomial coefficients for $((a + b)^{n}) : [Row n : (0 n), (1 n), (2 n), ..., (n n)]$

2.0History of Pascal's Triangle

Though the triangle was named after Blaise Pascal as a result of his work in the seventeenth century; it was known long before then. The triangle was referred to by the Indian mathematician Pingala (c. 200 BCE) as the "Meru Prastara" (Mount Meru). The Chinese mathematician Yang Hui (c. 13th century) also studied the triangle more extensively and so the triangle is referred to as "Yang Hui's Triangle" in China. In Persia, Omar Khayyam (c. 11th century) also studied its application for solving polynomial equations. However, Pascal is the first person to have formalized a lot of the triangles properties in his treatise, Traité du triangle arithmétique, published in or around 1653.

3.0Construction of Pascal's Triangle

The triangle starts with 1 at the top (row 0). Each number is obtained by adding the two numbers above it.

Example (first 6 rows):

$Row 0: 1$

$Row 1: 11$

$Row 2: 121$

$Row 3: 1331$

$Row 4: 14641$

$Row 5: 15101051$

General Rule:

The number at row n and position r is:

$(r n) = \frac{n !}{r ! ( n - r )!}$

4.0Properties of Pascal’s Triangle

Sum of Rows

The sum of the numbers in any row n is equal to 2n.

$Row 3: 1 + 3 + 3 + 1 = 8 = 2^{3}$
$Row 4: 1 + 4 + 6 + 4 + 1 = 16 = 2^{4}$

Symmetry

The triangle is perfectly symmetric. The numbers in each row are the same from left to right and right to left. This is because $(k n) = (n - k n)$

Hockey-Stick Identity

This property states that the sum of the numbers along a diagonal (starting from any 1 on the edge) is equal to the number below the last term in the next row, forming a "hockey stick" shape. Example: 1+3+6+10=20. In terms of combinations, $\sum_{i = r}^{n} (r i) = (r + 1 n + 1)$

Triangular and Square Numbers

The second diagonal (starting with 1, 3, 6, 10...) represents the triangular numbers. The sum of two consecutive triangular numbers equals a square number.

$1 + 3 = 4 = 2^{2}$

$3 + 6 = 9 = 3^{2}$

Fibonacci Sequence

By summing the numbers along the shallow diagonals of the triangle, you can find the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...).

Prime Number Property

If the second element in a row (after the first 1) is a prime number, then all the other numbers in that row are multiples of that prime number.

In Row 5 (1, 5, 10, 10, 5, 1), the number 5 is prime, and 10 is divisible by 5.

5.0How to Use Pascal’s Triangle?

Pascal’s Triangle is a versatile mathematical tool that can be used in numerous ways, especially for problems involving binomial expansions, combinations, and pattern recognition—key areas in JEE Mathematics. Here’s how you can effectively use Pascal’s Triangle in your problem-solving:

Finding Binomial Coefficients Quickly

Each entry in Pascal’s Triangle corresponds to a binomial coefficient $((k n))$ . For any binomial expansion $((a + b)^{n})$ , simply select the nth row of Pascal’s Triangle to obtain all coefficients instantly, without manual calculation.

Example:
The coefficients for $((x + y)^{4}) (ro w 4)$ are 1, 4, 6, 4, 1.

Expanding Binomial Expressions

Pascal’s Triangle provides the coefficients required for expanding powers of binomials using the Binomial Theorem.

$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k}$
Simply match the coefficients from the nth row to the respective terms in the expansion.

Solving Combinatorial Problems

In permutation and combination problems, the triangle gives the number of ways to choose (k) objects from (n) ((\binom{n}{k})). This is especially useful for probability and selection-based questions in JEE.

Example:
The number of ways to choose 3 out of 6 objects is found at the 6th row, 3rd entry: (\binom{6}{3} = 20).

Identifying and Using Patterns

Pascal’s Triangle reveals several mathematical patterns—like symmetry, the Hockey Stick identity, and Fibonacci numbers along certain diagonals—that can simplify complex algebraic or combinatorial problems.

Calculating Sums and Series

By using the properties of Pascal’s Triangle, you can quickly find the sum of combinations or powers of 2. For instance, the sum of the numbers in the nth row is always (2^n).

Applying in Probability

Coefficients from Pascal’s Triangle are used in binomial probability distributions to calculate probabilities of specific outcomes, such as in coin-tossing problems.

Verifying Algebraic Identities

Pascal’s Triangle aids in verifying and proving identities, such as the Binomial Theorem, combinatorial identities, and relationships among coefficients in polynomial equations.

6.0Solved Examples on Pascal’s Triangle

Example 1: Binomial Coefficient Calculation

Question:
Find the value of $((4 9))$ using Pascal’s Triangle.

Solution:
Locate the 9th row and the 4th position (starting from 0).

$(4 9) = \frac{9 !}{4 ! 5 !} = \frac{362880}{24 \times 120} = \frac{362880}{2880} = 126$

Example 2: Binomial Expansion Coefficient

Question:
What is the coefficient of $(x^{5})$ in the expansion of $((2 + x)^{7})$ ?

Solution:
The general term: $((k 7) 2^{7 - k} x^{k})$
For (k = 5):

$Coefficient = (5 7) \times 2^{7 - 5} = 21 \times 4 = 84$

Example 3: Application in Combinatorics

Question:
How many ways can a committee of 3 students be formed from a group of 8?

Solution:
Directly use Pascal’s Triangle: $((3 8))$

$(3 8) = \frac{8 !}{3 ! 5 !} = 56$

Example 4: Hockey Stick Identity

Question:
Verify the Hockey Stick Property for the numbers $((2 3) + (2 4) + (2 5))$ .

Solution:

$(2 3) + (2 4) + (2 5) = 3 + 6 + 10 = 19$
According to the Hockey Stick identity,

$(3 6) = 20$
But since the property includes all combinations from (n) to (n+k), we should check the sum:

$(2 3) + (2 4) + (2 5) = 3 + 6 + 10 = 19$
It differs by 1 from $((3 6))$ , so ensure the correct application: the sum from $((2 3)) t o ((2 6))$ yields $((3 7) = 35)$ , which matches for a complete diagonal.

Example 5: Sum of Row Elements

Question:
Calculate the sum of all numbers in the 8th row of Pascal’s Triangle.

Solution:
$S u m (= 2^{8} = 256)$ .

Example 6: Coefficient in a Mixed Term

Question:
Find the coefficient of $(a^{3} b^{2})$ in the expansion of $((a + b)^{5})$ .

Solution:
Use $((2 5) a^{3} b^{2})$ :

$(2 5) = 10$
So, the coefficient is 10.

Example 7: Recognizing Patterns

Question:
Show that the sum of the shallow diagonal of Pascal’s Triangle corresponds to a Fibonacci number. For the diagonal starting at $((0 4)), s u m ((0 4) + (1 3) + (2 2))$ .