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Binomial Theorem

Exercises in expanding powers of binomial expressions and finding specific coefficients.

Brackets Pascal's Triangle Level 1 Level 2 Exam-Style Description Help More Algebra

This is level 2: finding specific coefficients

1. If you expand the binomial expression \( (a + b)^5 \) the result would be:

\( {5 \choose 0} a^5 + {5 \choose 1} a^4 b + {5 \choose 2} a^3 b^2 + {5 \choose 3} a^2 b^3 + {5 \choose 4} ab^4 + {5 \choose 5} b^5 \)

Find the coefficient of the \( a^3 b^2 \) term

Correct Wrong

2. In the expansion of \( (c + d)^7 \) what is the coefficient of the \( c^3 d^4 \) term?

Correct Wrong

3. In the expansion of \( (e - f)^9 \) what is the coefficient of the \( e^2 f^7 \) term?

Correct Wrong

4. In the expansion of \( (2x + 3y)^4 \) what is the coefficient of the \( x^4 \) term?

Correct Wrong

5. In the expansion of \( (3x - 4y)^5 \) what is the coefficient of the \( y^5 \) term?

Correct Wrong

6. In the expansion of \( (2x + 3y)^4 \) what is the coefficient of the \( x^2y^2 \) term?

Correct Wrong

7. In the expansion of \( (3x - 4y)^5 \) what is the coefficient of the \( x^2y^3 \) term?

Correct Wrong

8. Find the the coefficient of the \(x^2\) term in the expansion of \( (x^2 + \frac{1}{x} )^7 \)

Correct Wrong

9. Find the constant term in the expansion of \( (2x^2 - \frac{1}{2x} )^6 \)

Correct Wrong

10. Use the expansion of \( (5 + x)^3 \) to find the exact value of \( 5.1^3 \).

Correct Wrong

This is Binomial Theorem level 2. You can also try:
Level 1


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Description of Levels

Brackets - The basics of expanding brackets

Pascal's Triangle - See how the coefficients are generated

Level 1 - Complete expansions of binomials

Level 2 - Finding specific coefficients

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Algebra including lesson Starters, visual aids, investigations and self-marking exercises.

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Binomial Theorem Revision

Using a Scientific Calculator

The function to work out a number in Pascal's Triangle appears as nCr where:

nCr Formula nCr button on calculator

Using a Graphic Display Calculator

With a GDC you can work out a whole row of Pascal's Triangle using the combinations function. Here is a screenshot of the TI Nspire.

Combinations function

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Typing Mathematical Notation

These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.

MathQuill Animation