Свойство 2. $$ C_n^m + C_n^{m + 1} = C_{n + 1}^{m + 1} ,\quad m = 0;1;2;...;n$$
Свойство 3. $$ C_n^0 + C_n^1 + ... + C_n^{n - 1} + C_n^n = 2^n$$
Свойство 4. $$ C_{2n}^0 + C_{2n}^2 + C_{2n}^4 + ... + C_{2n}^{2n} = C_{2n}^1 + C_{2n}^3 + C_{2n}^5 + ... + C_{2n}^{2n - 1} = 2^{2n - 1}$$
Свойство 5. $$C_{2n + 1}^0 + C_{2n + 1}^2 + C_{2n + 1}^4 + ... + C_{2n + 1}^{2n} = C_{2n + 1}^1 + C_{2n + 1}^3 + C_{2n + 1}^5 + ... + C_{2n + 1}^{2n + 1} = 2^{2n}$$
Свойство 6. $$C_n^{m + 1} = \frac{{n - m}}{{m + 1}}C_n^m$$