Properties of Binomial Coefficients

Properties of Binomial Coefficients

This lesson covers the following key concepts:

• Sum of all binomial coefficients
• Sum with alternate signs
• Sum of coefficients at even and odd positions
• Squares of binomial coefficients
• Products of binomial coefficients
• Pascal's identity and other relations
• Applications of differentiation and integration

Important Formulas

• $$C_0 + C_1 + C_2 + ... + C_n = 2^n$$
• $$C_0 - C_1 + C_2 - ... + (-1)^n C_n = 0$$
• $$C_0 + C_2 + C_4 + ... = C_1 + C_3 + ... = 2^{n-1}$$
• $$C_0^2 + C_1^2 + ... + C_n^2 = ^{2n}C_n$$
• $$C_1 + 2C_2 + 3C_3 + ... + nC_n = n \cdot 2^{n-1}$$
• $$\frac{C_0}{1} + \frac{C_1}{2} + ... + \frac{C_n}{n+1} = \frac{2^{n+1} - 1}{n+1}$$