Pascal's Triangle Practice Questions

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. It is named after the French mathematician Blaise Pascal, although it was studied by mathematicians in various cultures long before him.

Solved Examples

Example 1: Find the fifth row of Pascal’s triangle. 

Solution:

The Pascal triangle with 5 row is shown in the image below,

Example 2: Expand using Pascal Triangle (a + b)². 

Solution:

First write the generic expressions without the coefficients.  

 (a + b)² = c₀a²b⁰ + c₁a¹b¹ + c₂a⁰b² 

Now let’s build a Pascal’s triangle for 3 rows to find out the coefficients.  

The values of the last row give us the value of coefficients, c₀ = 1, c₁ = 2, c₂ =1

 (a + b)² = a²b⁰ + 2a¹b¹ + a⁰b² 

Thus verified. 

Example 3: Expand using Pascal Triangle (a + b)⁶. 

Solution:

First write the generic expressions without the coefficients.  

 (a + b)⁶ = c₀a⁶b⁰ + c₁a⁵b¹ + c₂a⁴b² + c₃a³b³ + c₄a²b⁴ + c₅a¹b⁵ + c₆a⁰b⁶  

Now let’s build a Pascal’s triangle for 7 rows to find out the coefficients.  

The values of the last row give us the value of coefficients. 

c₀ = 1, c₁ = 6, c₂ = 15, c₃ = 20, c₄ =15, c₅ = 6 and c₆ = 1. 

(a + b)⁶ = 1a⁶b⁰ + 6a⁵b¹ + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6a¹b⁵ + 1a⁰b⁶  

Example 4: Find the second element in the third row of Pascal’s triangle.

Solution:

To find the 2nd element in the 3rd row of Pascal’s triangle.

We know that the n^th row of Pascal’s triangle is ⁿC₀, ⁿC₁, ⁿC₂, ⁿC_3…

The Pascal Triangle Formula is, ⁿC_k = ^n-1C_k-1  + ^n-1C_k,where ⁿC_k represent (k+1)^th element in n^th row.

Thus, 2nd element in the 3rd row is,

³C₁ = ²C₀ + ²C₁
= 1 + 2
= 3

Thus, the second element in the third row of Pascal’s triangle is 3.

Practice Questions

Question 1: What is the sum of the elements in the fifth row of Pascal's triangle.

Question 2: Find the value of the 3rd element in the 7th row of Pascal's triangle without constructing the entire row.

Question 3: How can you use Pascal’s triangle to calculate the binomial coefficient \binom{6}{4}.

Question 4: If the sum of the elements in the n-th row of Pascal's triangle is 2ⁿ, find the sum of the elements in the 8th row.

Question 5: Determine the coefficient of a²b³ when expanding (a + b)⁵ using Pascal's Triangle.

Question 6: What is the difference between the 6th element in the 6th row and the 4th element in the 5th row of Pascal’s triangle.

Question 7: Using Pascal’s Triangle, expand the expression (x−y)⁴ and identify the coefficient of x²y².

Question 8: Find the sum of the diagonals in the first five rows of Pascal's Triangle.