State the expansion of (a+bx)^n.

a^n+\binom n1a^{n-1}bx+\binom n2a^{n-2}(bx)^2+\cdots+(bx)^n

5\times4\times3\times2\times1=120

How does Pascal's triangle give binomial coefficients?

Each entry is the sum of the two above it; row n gives \binom n0,\ldots,\binom nn.

How do you find the coefficient of x^k in a product of two brackets?

Multiply the matching terms from each expansion whose powers add to k, then add the products.

Estimate 1.03^8 using the first three terms.

(1+3x)^8 with x=0.01: 1+0.24+0.0252=1.2652

Why do binomial probabilities sum to 1?

They are the terms of ((1-p)+p)^n=1^n=1.

Binomial expansion for positive integer nAQA A-Level Maths: Flashcards

Card 1 of 130 of 13 known

Question

Formula for $\binom nr$?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

Formula for $\binom nr$ ?: $\frac{n!}{r!(n-r)!}$
Value of $\binom n0$ , $\binom n1$ , $\binom nn$ ?: $1$ , $n$ and $1$ .
$\binom nr$ equals which other coefficient?: $\binom n{n-r}$
State the expansion of $(a+bx)^n$ .: $a^n+\binom n1a^{n-1}bx+\binom n2a^{n-2}(bx)^2+\cdots+(bx)^n$
General term in $x^r$ of $(a+bx)^n$ ?: $\binom nr a^{n-r}b^rx^r$
How many terms are in the expansion of $(a+bx)^n$ ?: $n+1$
What is $5!$ ?: $5\times4\times3\times2\times1=120$
How does Pascal's triangle give binomial coefficients?: Each entry is the sum of the two above it; row $n$ gives $\binom n0,\ldots,\binom nn$ .
Coefficient of $x^2$ in $(2+x)^6$ ?: $\binom62\times2^4=240$
How do you find the coefficient of $x^k$ in a product of two brackets?: Multiply the matching terms from each expansion whose powers add to $k$ , then add the products.
Estimate $1.03^8$ using the first three terms.: $(1+3x)^8$ with $x=0.01$ : $1+0.24+0.0252=1.2652$
If $X\sim B(n,p)$ , what is $P(X=r)$ ?: $\binom nrp^r(1-p)^{n-r}$
Why do binomial probabilities sum to $1$ ?: They are the terms of $((1-p)+p)^n=1^n=1$ .

Exam questions on Binomial expansion for positive integer n

Consider the expansion of $(2+x)^6$ in ascending powers of $x$ .
Hence find the coefficient of $x^2$ in the expansion of $(1+x)(2+x)^6$ .2 marks
In the expansion of $(1+3x)^n$ , where $n$ is a positive integer, the term in $x$ is $24x$ .
Use the first three terms of the expansion to estimate the value of $1.03^8$ .2 marks
Consider the expansion of $(3-2x)^7$ in ascending powers of $x$ .
Find the first three terms in the expansion, in ascending powers of $x$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).