Binomial Theorem — NDA Maths

Formulas (8)

• The Binomial Theorem & the General Term · General term
  $$T_{r+1} = \binom{n}{r}\, a^{\,n-r} b^{\,r}$$
• Binomial Coefficients — C(n, r) · Binomial coefficient
  $$\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$$
• Finding a Specific Term or Coefficient · Set the exponent, solve for r
  $$\text{exponent of } x \text{ in } T_{r+1} = k \ \Rightarrow\ r \ \Rightarrow\ \text{coefficient}$$
• The Middle Term · Middle term, n even
  $$T_{\frac{n}{2}+1} = \binom{n}{n/2}\, a^{\,n/2} b^{\,n/2}$$
• The Term Independent of x (Constant Term) · Constant term condition
  $$\text{exponent of } x \text{ in } T_{r+1} = 0 \ \Rightarrow\ r$$
• Conditions Linking Coefficients · First-three-terms shape
  $$\binom{n}{1}a = (\text{2nd}),\quad \binom{n}{2}a^2 = (\text{3rd}) \ \Rightarrow\ \text{divide, solve } n$$
• Counting Terms in Products and Powers · Distinct terms of a trinomial power
  $$(a+b+c)^n \ \longrightarrow\ \binom{n+2}{2}\ \text{distinct terms}$$
• Rational Terms & the General-Index Series · Rational-term test
  $$\tfrac{n-r}{j} \in \mathbb{Z}\ \text{ and }\ \tfrac{r}{k} \in \mathbb{Z}$$

Watch out for (6)

• Term number is r + 1, not r
  → The Binomial Theorem & the General Term
• Equal coefficients gives TWO cases
  → Binomial Coefficients — C(n, r)
• Collect every power of x first
  → Finding a Specific Term or Coefficient
• Odd n has two middle terms
  → The Middle Term
• Multiply the bases before raising the power
  → Counting Terms in Products and Powers
• BOTH exponents must be integers, not just one
  → Rational Terms & the General-Index Series

Formulas (4)

• Sum of All Coefficients — Put x = 1 · Sum of coefficients = f(1)
  $$\sum_{r=0}^{n}\binom{n}{r} = 2^n, \qquad \sum_{r=0}^{n}\binom{n}{r}c^{r} = (1+c)^n$$
• Alternating & Odd/Even-Index Sums — Put x = −1 · Alternating sum and the split
  $$\sum_r (-1)^r \binom{n}{r} = 0, \qquad \text{even-sum} = \text{odd-sum} = 2^{\,n-1}$$
• Weighted Sums via Differentiation · Index-weighted sum
  $$\sum_{r=1}^{n} r\binom{n}{r} = n\,2^{\,n-1}$$
• Pascal's Rule & Coefficient Identities · Pascal's rule (applied twice)
  $$\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2} = \binom{n+2}{r}$$

Watch out for (3)

• Sum of coefficients uses x = 1, not x = 0
  → Sum of All Coefficients — Put x = 1
• Differentiate first, substitute second
  → Weighted Sums via Differentiation
• Pascal's rule needs adjacent lower indices on the SAME n
  → Pascal's Rule & Coefficient Identities

Formulas (2)

• The Conjugate Trick — Irrational Parts Cancel · Conjugate sum & product
  $$(a+\sqrt{b})^n + (a-\sqrt{b})^n \in \mathbb{Z}, \qquad (a+\sqrt{b})^n(a-\sqrt{b})^n = (a^2-b)^n$$
• Integer Part + Fractional Part · Fractional parts add to 1
  $$f + f' = 1, \qquad f' = (a-\sqrt{b})^n$$

Watch out for (2)

• Add the conjugate — don't expand the whole thing
  → The Conjugate Trick — Irrational Parts Cancel
• The conjugate power IS the missing fractional part
  → Integer Part + Fractional Part

Formulas (2)

• Remainders by the Binomial Trick · Base near a multiple of m
  $$(km \pm 1)^n \equiv (\pm 1)^n \pmod{m}$$
• Power of a Prime in n! (Legendre's Formula) · Legendre's formula
  $$E_p(n!) = \sum_{i\ge 1} \left\lfloor \dfrac{n}{p^{\,i}} \right\rfloor$$

Watch out for (2)

• Choose the base CLOSEST to a multiple of the divisor
  → Remainders by the Binomial Trick
• Count the prime, then divide by its exponent
  → Power of a Prime in n! (Legendre's Formula)