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1.4 Binomial Theorems and Combinatorics

Tags
Binomial theorem
Combination
Permutation
Pascal's triangle
Counting principle
Expansion
Factorial
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The binomial theorem: expansion of (a+b)n(a + b)^n, n∈Nn ∈ N. (1.9)
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Use of Pascal’s triangle and nCr^nC_r.
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Counting principles, including permutations and combinations. (AHL 1.10)
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Extension of the binomial theorem to fractional and negative indices, ie (a+b)n(a + b)^n, n∈Qn ∈ Q.

Use of Pascal’s triangle

Figure 1.4.1 Pascal’s triangle
For the binomial expansion of (ax+b)n(ax+b)^n where n∈Nn∈ N:
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The powers of ax decrease by 1, and on the other hand, the powers of b increases by 1
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ax(n−r)bra x^{(n-r)} b^r
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The expansion contains n+1n+1 terms
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The sum of the exponents of axax and bb in each term equals nn
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The coefficients of the terms correspond to the nnth row of Pascal’s triangle

Binomial series

Binomial expansion of two terms (ax+b)n(ax+b)^n for n∈Z+ n∈Z^+ is given by:
∑r=0n(nr)(ax)n−r(b)r=axn+(n1)axn−1b+…+(nr)axn−rbr+…+bn\sum_{r=0}^{n} \binom{n}{r} (ax)^{n-r} (b)^r = a x^n + \binom{n}{1} a x^{n-1} b + \ldots + \binom{n}{r} a x^{n-r} b^r + \ldots + b^n
(nr)\binom{n}{r} is called the binomial coefficient.
This can be extended to rational exponents (n∈Qn ∈ Q): (122)=12⋅(12−1)2!\binom{\frac12}{2} = \frac{\frac12 \cdot (\frac12-1)}{2!}
General term: (nr)axn−rbr\binom{n}{r} a x^{n-r} b^r, is used to find the coefficient of a single term in the expansion.

Binomial series HL

The binomial series is an extension of the binomial expansion. It is used for negative or fractional powers– for example:
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1(a+b)n=(a+b)−n\frac{1}{(a+b)^n} = (a+b)^{-n}
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(a+b)mn=(a+b)mn\sqrt[n]{(a+b)^m} = (a+b)^{\frac{m}{n}}
For such powers, the value for a must be changed to 1.
(a+b)n=an(1+n(ba)+n(n−1)2!(ba)2+…)(a+b)^n = a^n \left(1 + n \left(\frac{b}{a}\right) + \frac{n(n-1)}{2!} \left(\frac{b}{a}\right)^2 + \ldots \right), n∈Qn ∈ Q
*This is given in the formula booklet.
If a=1a=1 and b=xb=x, the expression can be simplified to:
(1+x)p=1+px+p(p−1)2!x2+p(p−1)(p−2)3!x3+…=∑n=0∞p(p−1)⋯(p−n+1)n!xn(1+x)^p = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 + \ldots = \sum_{n=0}^{\infty} \frac{p(p-1) \cdots (p - n + 1)}{n!}x^n
*This is not given in the formula booklet.
This formula is only valid for ∣x∣<1 |x|<1, which is called the interval of convergence.
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For an expansion (ax+b)n(ax+b)^n, the interval of convergence would be −ba<x<ba-\frac ba<x<\frac ba.

Counting Principles

The product principle states that if there are m distinct ways to perform one operation, and for each of these ways, there are n distinct ways to perform a second independent operation, then the total number of ways to perform both operations sequentially is m ✕ n.
The sum principle states that if there are choices for m or n way of performing an operation, then there are m + n different ways of performing either one of the two operations.
In most questions, the word and implies the product principle, while the word or suggests the sum principle.
There are a few combinatorics you need to be aware of:
Terminology
Definition
Factorial
n!=n(n−1)(n−2)(n−3)...3⋅2⋅1n! = n(n-1)(n-2)(n-3)...3 \cdot 2\cdot 1, for n≥1n \geq 1 n!n! is the number of possible arrangements of nn items. The number of arrangements with kk repeated elements is n!k!\frac {n!}{k!} Note that 0!=1 0!=1 A property of factorials: n!=n⋅(n−1)!⇒n!(n−1)!=nn!=n\cdot(n-1)! ⇒ \frac{n!}{(n-1)!}=n
Permutation 
An arrangement of objects in a definite order. Picking rr elements with consideration of order from nn elements: nPr=n!(n−r)!_nP_r=\frac{n!}{(n-r)!}
Combination
A selection of objects where the order or arrangement does not matter: nCr=(nr)=n!r!(n−r)!_nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}
1.4 Binomial theorem and combinatorics
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1.4 Binomial Theorems and Combinatorics