The Binomial Theorem and Pascal’s Triangle

Definition A binomial is a sum of two terms, $a + b$. For an integer $n \ge 1$ the power $(a + b)^n$ is the product of $n$ factors equal to $a + b$, and $(a + b)^0 = 1$. Treating $a$ and $b$ as two separate variables, multiplying out and collecting like terms gives a sum of terms of the form $c\,a^{n-k} b^k$ for $k = 0, 1, \dots, n$, so the result has $n + 1$ terms and the exponents in each term add up to $n$. The binomial theorem below is about these nonnegative integer exponents $n = 0, 1, 2, \dots$, and the last section shows what changes for other exponents.

Definition Let $n \ge 0$ and $k$ be integers with $0 \le k \le n$. A combination of $k$ objects from $n$ different objects is a choice of $k$ of them in which order does not matter, in other words a subset with $k$ elements. The number of such combinations is written

read as $n$ choose $k$, and is also written $C(n, k)$. There is exactly one way to choose nothing and one way to choose everything, so $\binom{n}{0} = \binom{n}{n} = 1$.

Definition For integers $n \ge 0$ and $0 \le k \le n$,

where $0! = 1$. To see why, count ordered choices first. The number of ways to pick $k$ objects one after another is

Each set of $k$ objects turns up in $k!$ different orders, so dividing by $k!$ counts each set exactly once. After cancelling, for $k \ge 1$,

with $k$ factors on top. The formula also shows the symmetry $\binom{n}{k} = \binom{n}{n - k}$.

Definition The numbers $\binom{n}{k}$ are called binomial coefficients because they are exactly the coefficients in the expansion of $(a + b)^n$. The coefficient of $a^{n-k} b^k$ is $\binom{n}{k}$. The reason is counting. Multiplying out

means choosing $a$ or $b$ from each of the $n$ factors, multiplying the choices, and adding up all the results. A product equals $a^{n-k} b^k$ exactly when $b$ was chosen from $k$ of the factors, and there are $\binom{n}{k}$ ways to choose those $k$ factors.

Definition Pascal’s triangle is the triangular array whose row $n$, for $n = 0, 1, 2, \dots$, lists the binomial coefficients $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$. Each row starts and ends with $1$, and each entry in between is the sum of the two entries just above it. This is Pascal’s rule, which holds for integers $1 \le k \le n$,

Counting explains it. Mark one of $n + 1$ objects. A subset of $k$ of them either leaves out the marked object, and there are $\binom{n}{k}$ such subsets, or contains it, and there are $\binom{n}{k - 1}$ ways to choose the rest.

Definition For all real numbers $a$ and $b$ and every integer $n \ge 0$,

with the convention that $x^0 = 1$ for every $x$, including $x = 0$. The lesson on power functions left $0^0$ undefined, and inside this formula it simply means that a factor with exponent $0$ is left out. The term with index $k$ is $\binom{n}{k} a^{n-k} b^k$, so a single term can be found without writing out the others. The theorem can be proved by mathematical induction on $n$, with Pascal’s rule doing the work in the inductive step.

Definition For an integer exponent $n \ge 0$ the expansion of $(1 + x)^n$ has exactly $n + 1$ terms and holds for every real $x$. For a real exponent $\alpha$ the same pattern of coefficients is

When $\alpha = n$ is a nonnegative integer, every coefficient with $k > n$ contains the factor $n - n = 0$, which is why the expansion stops. When $\alpha$ is not a nonnegative integer, such as $-1$ or $\frac{1}{2}$, no coefficient is zero, and the expansion becomes the infinite binomial series

which converges to $(1 + x)^\alpha$ for $\lvert x \rvert < 1$ and diverges for $\lvert x \rvert > 1$. The infinite version belongs to calculus, in the lesson on the binomial series.