Binomial coefficients & unknowns
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Flip to reveal answersGeneral term of (a + b)ⁿ?
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Question
General term of (a + b)ⁿ?
Answer
T_(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ — choose r of the b's.
Question
How do you find the coefficient of a specific power xᵏ?
Answer
Use the general term; set the power of x equal to k to fix r, then compute ⁿCᵣ aⁿ⁻ʳ bʳ.
Question
How do you find an unknown constant from a given coefficient?
Answer
Write the coefficient as an expression in the unknown, set it equal to the given number, and solve.
Question
Coefficient of x² in (1 + ax)ⁿ (general term)?
Answer
ⁿC₂ a².
Question
Coefficient of x³ in (2 + x)⁵?
Answer
⁵C₃ × 2² = 10 × 4 = 40.
Question
In (1 + ax)⁵ the coefficient of x² is 40 (a > 0). Find a.
Answer
10a² = 40 ⇒ a² = 4 ⇒ a = 2.
Question
Common mistake finding a binomial coefficient?
Answer
Forgetting to raise the inside coefficient to the power r — (2x)² = 4x², not 2x².
Question
Why must you include the ⁿCᵣ?
Answer
The coefficient counts the ⁿCᵣ ways that term is formed — the powers alone aren't enough.
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Full study notes for Binomial coefficients & unknowns
Topic 1.10 hub
Counting & binomial (HL only)
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