Binomial coefficients
Select the required term without expanding every part of a power.
What you will learn
- Use combinations to identify a term.
- Distinguish a coefficient from the full term.
Why combinations appear
In (a+b)ⁿ, each factor contributes either a or b. A term with r copies of b arises by choosing those r factors from n, giving the multiplier C(n,r). The remaining n−r factors contribute a.
For a positive integer n, the expansion has n+1 terms. To find the coefficient of a particular power of x, identify r and include every numerical factor attached to x.
Keep the whole term together
In (a+bx)ⁿ the term containing xʳ is C(n,r)aⁿ⁻ʳbʳxʳ. The coefficient omits xʳ but includes bʳ. A negative b changes the sign of odd-powered terms.
Worked example
Find the coefficient of x² in (3+2x)⁴.
Show the worked solution
- Choose r=2, so C(4,2)=6.
- The remaining constant factor is 3²=9.
- Include 2² from (2x)²: 6×9×4=216.
Answer 216
Common mistakes
- Do not forget to raise the coefficient of x to the selected power.
- The coefficient is a number, not a number followed by x².
What can you explain now?
Find the coefficient of x in (2−x)³.
Compare with the explanation
−12
The linear term is C(3,1)×2²×(−x)=−12x.
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These lessons teach the topics represented in our current practice sets. They are not a complete course for every paper or option in the qualification.
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