Binomial expansion

​​In a nutshell

The binomial expansion formula allows you to expand $$(x+y)^n$$​ quickly. The coefficients of the terms in this expansion are given by entries of Pascal's triangle, and these can be quickly computed using $$^nC_r = \binom{n}{r} = \dfrac{n!}{r!(n-r)!}$$​.

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The binomial expansion

The binomial expansion is given by:

​$$(x+y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + \dots + \binom{n}{r}x^{n-r}y^r + \dots + \binom{n}{n-1}xy^{n-1} + \binom{n}{n}y^n$$​​

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Recall that $$\binom{n}{r} =\ ^nC_r$$​:

In order to compute the coefficient of the $$x^{n-r}y^r$$ term, you need to count the number of ways of choosing $$r$$​ copies of $$y$$​ from the set of $$n$$ brackets, and there are $$\binom{n}{r}$$​ ways to do so. Equivalently, you need to count the number of ways of choosing $$n-r$$​ copies of $$x$$​ from the $$n$$ brackets, and there are $$\binom{n}{n-r}$$ ways to do so. This proves that $$\binom{n}{r} = \binom{n}{n-r}$$.

Example 1

What is the expansion of $$(x+y)^7$$?

Substitute $$n=7$$ into the above formula:

​$$(x+y)^7 = \binom{7}{0}x^7 + \binom{7}{1}x^6y + \binom{7}{2}x^5y^2 + \binom{7}{3}x^4y^3 + \binom{7}{4}x^3y^4 + \binom{7}{5}x^2y^5 + \binom{7}{6}xy^6 + \binom{7}{7}y^7$$​​

Compute the coefficients $$^7C_r$$ (either using factorial notation, or by using the $$^nC_r$$​ button on your calculator, or by using Pascal's triangle):

​​​$$\begin{aligned}\binom{7}{0} &= \dfrac{7!}{0! \times 7!}&= 1 \\\binom{7}{1} &= \dfrac{7!}{1! \times 6!}&= 7 \\\binom{7}{2} &= \dfrac{7!}{2! \times 5!}&= 21 \\\binom{7}{3} &= \dfrac{7!}{3! \times 4!}&= 35 \\\binom{7}{4} &= \dfrac{7!}{4! \times 3!}&= 35 \\\binom{7}{5} &= \dfrac{7!}{5! \times 2!}&= 21 \\\binom{7}{6} &= \dfrac{7!}{6! \times 1!}&= 7 \\\binom{7}{7} &= \dfrac{7!}{7! \times 0!}&= 1 \\\end{aligned}$$​​

Therefore, $$\underline{(x+y)^7 = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^4 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7}$$.​​

Ascending powers

Typically when expanding a binomial of the form $$(a + bx)^n$$​, where $$a$$​ and $$b$$​ are constants and $$x$$​ is a variable, you should write the terms so that the degree of $$x$$​ is increasing from left to right. This is referred to as writing the expansion in ascending powers of $$x$$​.

Example 2

What is the expansion of $$(4-3t)^5$$?

The binomial expansion tells you that:

$$(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$$

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Substitute $$x = 4, y = -3t$$ to get:

​$$\begin{aligned}(4-3t)^5 &= 4^5 + 5 \times 4^4 \times (-3t) + 10 \times 4^3 \times (-3t)^2 + 10 \times 4^2 \times (-3t)^3 + 5 \times 4 \times (-3t)^4 +(-3t)^5 \\ &= 1024 -3840t + 5760t^2 -4320t^3 + 1620t^4 - 243t^5\end{aligned}$$

Therefore, $$\underline{ (4-3t)^5 = 1024 - 3840t + 5760t^2 - 4320t^3 + 1620t^4 -243t^5}$$.​​​