Binomial Theorem — JEE Mains Mathematics

Formulas (3)

• Setting the power of x · General term
  $$T_{r+1}=\binom nr\,a^{\,n-r}\,b^{\,r}$$
• Simplify the bracket first · A common collapse
  $$\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-\sqrt x}=x^{1/3}-x^{-1/2}$$
• Equating coefficients from two expansions · Symmetry of coefficients
  $$\binom nr=\binom n{n-r}$$

Watch out for (3)

• T with index r + 1
  → Setting the power of x
• Check the numerator's sign
  → Simplify the bracket first
• Signs from the second term
  → Equating coefficients from two expansions

Formulas (2)

• The ratio of neighbouring coefficients · Neighbour ratio
  $$\frac{\binom nr}{\binom n{r-1}}=\frac{n-r+1}{r}$$
• Middle, greatest and end terms · Start versus end
  $$\frac{T_k\ \text{(from start)}}{T_k\ \text{(from end)}}=\left(\frac ab\right)^{n-2k+2}$$

Watch out for (2)

• Which r is which
  → The ratio of neighbouring coefficients
• The exponent is n - 2k + 2
  → Middle, greatest and end terms

Formulas (2)

• Coefficients in a product · Two factors
  $$(1+x)^p(1-x)^q:\quad[x]=p-q,\quad[x^2]=\tfrac{(p-q)^2-(p+q)}{2}$$
• Three-term brackets · Multinomial term
  $$(p+q+r)^n:\ \ \frac{n!}{a!\,b!\,c!}\,p^a\,q^b\,r^c,\quad a+b+c=n$$

Watch out for (2)

• Include every combination
  → Coefficients in a product
• List every triple
  → Three-term brackets

Formulas (2)

• Counting and adding rational terms · Rational-term condition
  $$\binom nr p^{\frac{n-r}{a}}q^{\frac rb}\ \text{is rational exactly when}\ a\mid(n-r)\ \text{and}\ b\mid r$$
• Integral parts through the conjugate · Conjugate pair
  $$x=(a+\sqrt b)^n,\ y=(a-\sqrt b)^n:\quad x+y\in\mathbb Z,\ \ 0<y<1\Rightarrow[x]=x+y-1$$

Watch out for (2)

• Both exponents must be whole
  → Counting and adding rational terms
• A negative conjugate
  → Integral parts through the conjugate

Formulas (2)

• Substituting x = 1, −1 and other values · Even and odd parts
  $$\sum_{r\ \text{even}}a_r=\frac{f(1)+f(-1)}{2},\qquad\sum_{r\ \text{odd}}a_r=\frac{f(1)-f(-1)}{2}$$
• Weighted sums: r C(n, r) and r² C(n, r) · Weighted sums
  $$\sum_{r}r\binom nr=n\,2^{n-1},\qquad\sum_{r}r^2\binom nr=n(n+1)\,2^{n-2}$$

Watch out for (2)

• Remove the terms outside the range
  → Substituting x = 1, −1 and other values
• r² is not r · r in the identity
  → Weighted sums: r C(n, r) and r² C(n, r)

Formulas (2)

• Coefficients divided by r + 1 · Absorption into n + 1
  $$\sum_{r=0}^{n}\frac{\binom nr}{r+1}=\frac{1}{n+1}\sum_{r=0}^{n}\binom{n+1}{r+1}=\frac{2^{n+1}-1}{n+1}$$
• Products of coefficients: Vandermonde · Vandermonde's identity
  $$\sum_{r}\binom mr\binom n{k-r}=\binom{m+n}{k}$$

Watch out for (2)

• Trim the ends you are not given
  → Coefficients divided by r + 1
• Match the indices before summing
  → Products of coefficients: Vandermonde

Formulas (2)

• The hockey-stick identity · Hockey stick
  $$\sum_{k=r}^{n}\binom kr=\binom{n+1}{r+1}$$
• Summing a geometric series of expansions · Collapsing the sum
  $$\sum_{k=0}^{n}x^k(1+x)^{n-k}=(1+x)^{n+1}-x^{n+1}$$

Watch out for (2)

• Start the column at the right row
  → The hockey-stick identity
• The top power goes up by one
  → Summing a geometric series of expansions

Formulas (2)

• Remainders of large powers · Base near a multiple
  $$(mq\pm1)^n=m(\dots)+(\pm1)^n$$
• What an expression is divisible by · Removing the first two terms
  $$(1+m)^n-mn-1=m^2\left[\binom n2+\binom n3m+\dots\right]$$

Watch out for (2)

• Leftover factors
  → Remainders of large powers
• Pair the terms to match the signs
  → What an expression is divisible by