Properties of Binomial Coefficients
Binomial coefficients satisfy symmetry, Pascal's rule, and summation identities such as the sum of squares of coefficients equalling 2nCn.
What is Properties of Binomial Coefficients?
Binomial coefficients satisfy symmetry, Pascal's rule, and summation identities such as the sum of squares of coefficients equalling 2nCn.
Key formula / rule: Middle term is the greatest coefficient for a positive integral index.
Key points
- Derive coefficient identities by substitution into the expansion.
- Evaluate the greatest term or greatest coefficient in an expansion.
Common exam trap
Applying positive-index identities to a fractional or negative index.
Definitions
- Term
Properties of Binomial Coefficients
- Meaning
Binomial coefficients satisfy symmetry, Pascal's rule, and summation identities such as the sum of squares of coefficients equalling 2nCn.
- Term
Properties of Binomial Coefficients — explanation
- Meaning
Substituting particular values of x and y, or differentiating the expansion, generates identities that would be painful to prove term by term. Recognising which substitution to make is the skill being tested.
Learning objectives
Derive coefficient identities by substitution into the expansion.
Evaluate the greatest term or greatest coefficient in an expansion.
Formulae
- Key point
Middle term is the greatest coefficient for a positive integral index.
- Key point
Differentiating the expansion produces identities involving r * nCr.
- Key point
Sum of coefficients of even positions equals that of odd positions, both 2^(n-1).
Prerequisites
MAT-U1-PC-T1-S2-C1
Common mistakes
Applying positive-index identities to a fractional or negative index.
Taking the middle term as a single term when n is odd.
Keywords
Properties
Binomial
Coefficients
Practice preview
If C(n, r) denotes the binomial coefficient nCr, which of the following statements is always true?…
easy
According to Pascal's Rule, C(n, r) + C(n, r-1) is equal to:…
easy
The value of C(m, 0)C(n, k) + C(m, 1)C(n, k-1) + C(m, 2)C(n, k-2) + ... + C(m, k)C(n, 0) is equal to:…
hard