Maths Extension 1 Year 11 - Module 4 - Lesson 7

Properties of Combinations & Pascal's Identity

Use this worksheet after reading the lesson to practise the key ideas and prove you can meet the success criteria.

Name
Date
Class

1. Lesson focus

Combination properties and Pascal identity. Use symmetry nCr=nC(n−r) and Pascal's identity nCr=(n−1)C(r−1)+(n−1)Cr.

Success criteria

  • Apply the lesson method accurately in a routine example.
  • Show each algebraic or domain step and check the result.
  • Explain why the method and restrictions fit the question.

2. Worked practice

1. Evaluate 8C3 and use symmetry to state 8C5.

2. Use Pascal's identity to evaluate 7C2+7C3.

3. Find the sum of the entries in row 5 of Pascal's triangle, starting row 0.

3. Challenge

Use Pascal's identity to find 9C4 from entries in row 8.

4. Multiple choice

By symmetry, 9C2 equals which combination?

A. 9C2+1

B. 9C7

C. 8C2

D. 9P7

Worked answer key

Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.

1. Evaluate 8C3 and use symmetry to state 8C5.

8C3=56; by symmetry 8C5=56.

2. Use Pascal's identity to evaluate 7C2+7C3.

7C2=21 and 7C3=35; their sum is 56=8C3.

3. Find the sum of the entries in row 5 of Pascal's triangle, starting row 0.

The row sum is 2⁵=32; entries 1,5,10,10,5,1 sum to 32.

Challenge. Use Pascal's identity to find 9C4 from entries in row 8.

9C4=8C3+8C4=56+70=126.

Multiple choice. By symmetry, 9C2 equals which combination?

Correct answer: B. For fixed n, nCr=nC(n−r), so 9C2=9C7.

HSCScience - Printable lesson worksheet - Maths Extension 1 Year 11 - Module 4 - Lesson 7

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