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NSWMaths Extension 1Quick questions

Combinatorics (ME-A1)

Quick questions on The binomial theorem and Pascal's triangle: expansion of (a+b)n(a + b)^n and the general term

15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the binomial theorem?
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For any non-negative integer nn,
What is expanded form for small nn?
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$(a+b)2=a2+2ab+b2,(a + b)^2 = a^2 + 2 a b + b^2,$
What is the general term?
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The (k+1)(k + 1)-th term in the expansion of (a+b)n(a + b)^n is
What is pascal's triangle?
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Each row of Pascal's triangle gives the coefficients of (a+b)n(a + b)^n for that nn. The entries on the edges are 11; each interior entry is the sum of the two above it (Pascal's rule (nk)=(nβˆ’1kβˆ’1)+(nβˆ’1k)\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}).
What is sum identities?
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βˆ‘k=0n(nk)=2n\sum_{k = 0}^{n} \binom{n}{k} = 2^n. (Set a=b=1a = b = 1.)
What is finding specific terms?
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Coefficient of xkx^k in (ax+b)n(a x + b)^n: (nk)akbnβˆ’k\binom{n}{k} a^k b^{n - k} (chosen so the xx-power is kk).
What is coefficient of x3x^3 in (1+2x)6(1 + 2 x)^6?
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General term: Tk+1=(6k)(2x)k=(6k)2kxkT_{k + 1} = \binom{6}{k} (2 x)^k = \binom{6}{k} 2^k x^k.
What is independent term?
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Find the term independent of xx in (2x2+1x)6\left( 2 x^2 + \frac{1}{x} \right)^6.
What is sum identity?
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Prove βˆ‘k=0n(nk)=2n\sum_{k = 0}^{n} \binom{n}{k} = 2^n using the binomial theorem.
What is approximate (1.01)5(1.01)^5?
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Use (1+x)5β‰ˆ1+5x+10x2(1 + x)^5 \approx 1 + 5 x + 10 x^2 for small xx, with x=0.01x = 0.01.
What is negative-term expansion?
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(xβˆ’1)4=x4βˆ’4x3+6x2βˆ’4x+1(x - 1)^4 = x^4 - 4 x^3 + 6 x^2 - 4 x + 1 (signs alternate because b=βˆ’1b = -1). :::
What is coefficient of xkx^k in n ^n?
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(nk)akbnβˆ’k\binom{n}{k} a^k b^{n - k} (chosen so the xx-power is kk).
What is term independent of xx?
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set the power of xx in Tk+1T_{k + 1} to 00 and solve for kk.
What is wrong power balance?
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In (axp+bxq)n(a x^p + b x^q)^n, the power of xx in Tk+1T_{k + 1} is p(nβˆ’k)+qkp(n - k) + q k, not pk+q(nβˆ’k)p k + q(n - k). Track carefully.
What is forgetting anβˆ’ka^{n - k} when aβ‰ 1a \neq 1?
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(2x+3)5(2 x + 3)^5 has Tk+1=(5k)(2x)5βˆ’k(3)kT_{k + 1} = \binom{5}{k} (2 x)^{5 - k} (3)^k. Both 22 and 33 need their powers.

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