What is \binom nr in words?

The number of ways to choose r items from n.

How is Pascal's triangle built?

Each entry is the sum of the two entries above it; row n gives the coefficients of (1+x)^n.

Rule linking adjacent coefficients?

\binom nr+\binom n{r+1}=\binom{n+1}{r+1}

Expansion of (a+bx)^n?

a^n+\binom n1a^{n-1}bx+\binom n2a^{n-2}b^2x^2+\ldots+b^nx^n

How do you find the coefficient of x^2 in a product of two brackets?

Add every pair of terms, one from each bracket, whose powers sum to 2.

How would you use (1+2x)^5 to estimate 1.02^5?

Take x=0.01 and use the first few terms: 1+10x+40x^2+\ldots

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

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Formula for $\binom nr$?

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Formula for $\binom nr$ ?: $\frac{n!}{r!\,(n-r)!}$
What is $\binom nr$ in words?: The number of ways to choose $r$ items from $n$ .
What are $\binom n0$ and $\binom nn$ ?: Both equal $1$ .
Symmetry of binomial coefficients?: $\binom nr=\binom n{n-r}$
How is Pascal's triangle built?: Each entry is the sum of the two entries above it; row $n$ gives the coefficients of $(1+x)^n$ .
Rule linking adjacent coefficients?: $\binom nr+\binom n{r+1}=\binom{n+1}{r+1}$
Expansion of $(a+bx)^n$ ?: $a^n+\binom n1a^{n-1}bx+\binom n2a^{n-2}b^2x^2+\ldots+b^nx^n$
General term in $x^r$ of $(a+bx)^n$ ?: $\binom nr\,a^{n-r}b^r\,x^r$
Expand $(1+x)^4$ .: $1+4x+6x^2+4x^3+x^4$
Formula for $P(X=r)$ when $X\sim B(n,p)$ ?: $\binom nrp^r(1-p)^{n-r}$
How do you find the coefficient of $x^2$ in a product of two brackets?: Add every pair of terms, one from each bracket, whose powers sum to $2$ .
How would you use $(1+2x)^5$ to estimate $1.02^5$ ?: Take $x=0.01$ and use the first few terms: $1+10x+40x^2+\ldots$

Exam questions on Binomial expansion for positive integer powers

The expression $(1+2x)^5$ is expanded in ascending powers of $x$ .
Use the first three terms of the expansion, with a suitable value of $x$ , to estimate $1.02^5$ .2 marks
The random variable $X$ is the number of heads obtained when a fair coin is tossed 6 times, so $X\sim B\left(6,\frac12\right)$ .
Without evaluating either probability, explain why $P(X=4)=P(X=2)$ .2 marks
The expression $(2+kx)^5$ , where $k$ is a positive constant, is expanded in ascending powers of $x$ . The coefficient of $x^2$ in the expansion is $720$ .
Find the value of $k$ .3 marks

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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).