Binomial Coefficients

Binomial coefficients $(k n)$ are the coefficients in the expansion of powers of binomials. They link combinatorics with algebra.

The Binomial Theorem

Theorem (Binomial Theorem)

For any non-negative integer

n

(x + y)^{n} = k = 0 \sum n (k n) x^{n - k} y^{k}

(x + y)^{n} = (0 n) x^{n} + (1 n) x^{n - 1} y + \dots + (n n) y^{n}

[Image of Pascal's Triangle]

The coefficients are the entries in Pascal's Triangle. Each entry is the sum of the two numbers directly above it.

Key Identities

Sum of Rows: $\sum_{k = 0}^{n} (k n) = 2^{n}$ (Set $x = 1, y = 1$ ).

Alternating Sum: $\sum_{k = 0}^{n} (- 1)^{k} (k n) = 0$ (Set $x = 1, y = - 1$ ).

$(0 n) + (2 n) + \dots = (1 n) + (3 n) + \dots = 2^{n - 1}$ .

Hockey Stick Identity:

i = r \sum n (r i) = (r + 1 n + 1)

Practice Problems

Exercise (Problem 1)

Find the coefficient of

x^{3} y^{2}

in the expansion of

(2 x - 3 y)^{5}

Exercise (Problem 2)

Compute the sum:

1 \cdot (1 n) + 2 \cdot (2 n) + \dots + n \cdot (n n)

(Hint: Take the derivative of $(1 + x)^{n}$ )

Exercise (Problem 3)

Prove that

(n 2 n)

is always even for

n \geq 1