j-james/math

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All my math notes, now in Markdown.

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Binomial Theorem

Learning Targets

You should be able to

Concepts / Definitions

Expanding a Binomial

$(a + b)^0 = 1a^0$
$(a + b)^1 = 1a^1 + 1b^1$
$(a + b)^2 = 1a^2 + 2ab + 1b^2$
$(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3$
$(a + b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4$
$(a + b)^5 = 1a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + 1b^5$

Pascal’s triangle corresponds to a value of $_nC_r$

\[_0C_0\] \[_1C_0\ _1C_1\] \[_2C_0\ _2C_1\ _2C_2\] \[_3C_0\ _3C_1\ _3C_2\ _3C_3\] \[_4C_0\ _4C_1\ _4C_2\ _4C_3\ _4C_4\] \[_5C_0\ _5C_1\ _5C_2\ _5C_3\ _5C_4\ _5C_5\]

Binomial Theorem

\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] \[(a + b)^n = nC_0 a^n b^0 + nC_1 a^{n-1} b^1 + nC_2 a^{n-2} b^2 + ... + nC_{n-1} a^1 b^{n-1} + nC_n a^0 b^n\]

Binomial Theorem / Pascal’s Triangle uses

Head and Tail Categories Powers of Elevens Number Patterns in diagonals Shallow Diagonals

Exercises

  1. Evaluate $\binom 73$ without calculator
  2. Use the Binomial Theorem to expand $(2x + y)^4$ (no calculator)
  3. Use the Binomial Theorem to expand $(\sqrt x - \sqrt y)^6$ (calculator okay)
  4. Use the Binomial Theorem to expand $(x^{-2} + 3)^5$
  5. Find the coefficient of the $x^{11}y^3$ term for $(x+y)^{14}$
  6. Find the sixth term of $(x - 2)^6$
  7. If $n$ is a positive integer, show that $\binom n0 + \binom n1 + … + \binom nn = 2^n$