Worksheet: Pascal’s Triangle

In this worksheet, we will learn how to solve problems on Pascal’s triangle.

Q1:

In the picture of Pascal’s triangle shown, what type of numbers appear in the diagonal indicated by the arrow?

  • AThe Fibonacci numbers
  • BThe cube numbers
  • CThe square numbers
  • DThe triangular numbers

Q2:

What is the sum of the terms in the 30th row of Pascal’s triangle?

  • A21
  • B2
  • C2
  • D21
  • E2

Q3:

The figure shows a section of Pascal’s triangle. Without using a calculator, find the sum of the highlighted elements.

Q4:

What is sum of every other element in the 1,000th row of Pascal’s triangle?

  • A2
  • B2
  • C21
  • D2
  • E2

Q5:

What is the second element in the 500th row of Pascal’s triangle?

Q6:

The figure shows 7 adjacent elements in Pascal’s triangle.

Given that 5,985 is the 18th element on the 22nd row, find 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑓.

  • A𝑎=20,349, 𝑏=4,845, 𝑐=1,140, 𝑑=1,330, 𝑒=7,315, 𝑓=26,334
  • B𝑎=21,546, 𝑏=4,845, 𝑐=1,140, 𝑑=1,260, 𝑒=7,245, 𝑓=27,531
  • C𝑎=20,349, 𝑏=4,560, 𝑐=1,425, 𝑑=1,330, 𝑒=7,315, 𝑓=26,334
  • D𝑎=20,349, 𝑏=5,130, 𝑐=855, 𝑑=1,330, 𝑒=7,315, 𝑓=26,334
  • E𝑎=21,546, 𝑏=5,130, 𝑐=855, 𝑑=1,260, 𝑒=7,245, 𝑓=27,531

Q7:

William is playing a game where a ball is dropped onto an array of pegs, from vertically above the top peg, and it bounces down to numbered buckets at the bottom.

He only gets a prize if it falls into buckets 3 or 7. Find the probability that he gets a prize given that there is an even probability it will fall to the left or the right of any given peg.

  • A764
  • B732
  • C716
  • D7128
  • E7256

Q8:

In Pascal’s triangle, each entry is the sum of the two entries above it. In which row of Pascal’s triangle do three consecutive entries occur that are in the ratio 345?

Q9:

You have 7 pool balls. Find, using Pascal’s triangle, how many different ways you could choose just 3 of them, ignoring the order that you select them in.

Q10:

Use Pascal’s triangle to find the binomial expansion of (𝑥+𝑦).

  • A𝑥+4𝑥𝑦+6𝑥𝑦+4𝑥𝑦+𝑦
  • B2𝑥+4𝑥𝑦+6𝑥𝑦+4𝑥𝑦+2𝑦
  • C𝑥+4𝑥𝑦6𝑥𝑦+4𝑥𝑦
  • D𝑥4𝑥𝑦6𝑥𝑦4𝑥𝑦
  • E𝑥4𝑥𝑦6𝑥𝑦4𝑥𝑦+𝑦

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