The Binomial Theorem
- The binomial theorem is a method for expanding powers of binomials.
- For any positive integer $n$, it gives the expansion of $(a+b)^n$ as a sum of terms involving powers of $a$ and $b$.
Pascal's Triangle and nCr Notation
Pascal's Triangle
- Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.
- Each number is the sum of the two numbers directly above it.
- The numbers in Pascal's triangle correspond to the coefficients in the binomial expansion.
- The nth row (starting with row 0 at the top) gives the coefficients of the expansion of $(a+b)^n$.
- It's worth it memorizing the first few rows of Pascal's triangle, up to the 6th row (1, 5, 10, 10, 5, 1).
- This will help you expand any binomial power up to $(a + b)^5$.
We'll talk about Pascal's triangle later in the section on calculating binomial coefficients.
nCr Notation
- The notation $^nC_r$ (or $\binom{n}{r}$) represents the number of ways to choose $r$ items from a set of $n$ items, where order doesn't matter.
- It's read as "n choose r".
- The formula for the number of combinations is calculated as: $$ nCr = \frac{n!}{r!(n-r)!} $$
- For example, let's take a set of four items, A, B, C, and D.
- If we want to choose two items from this set, these are the possible combinations:
- A, B
- A, C
- A, D
- B, C
- B, D
- C, D
- There are six possible combinations, so we say that $^4C_2 = 6$.
- There are many alternative notations for $^nC_r$, such as $^n_rC$, $nCr$, $C^n_r$, and $\binom{n}{r}$.
- All of these are interchangeable, so use whichever one you prefer.
Example question
Calculate $^5C_2$.
Solution
$$^5C_2 = \frac{5!}{2!(5-2)!} = \frac{5 \cdot 4}{2 \cdot 1} = 10$$
This means there are 10 ways to choose 2 items from a set of 5.
When calculating $^nC_r$, it's often easier to cancel out common factors in the numerator and denominator before multiplying.
$^nC_r$ also corresponds to Pascal's triangle, and is equal to the $r$th number on the $n$th row, starting counting from $0$.
