other_note_coefficients

Binomial Coefficient

• [{n \choose r} = \frac{ n!}{ r!(n - r)!}] for [ 0 \le r \le n]
  • Factorial Formula

It is the polynomial expansion of the binomial power [(1 + x)^n]

[{n \choose r}] represents the number of possible combinations of [n] objects taken [r] at a time.

Combinatorial identities

[{n \choose r} = { n - 1 \choose r - 1} + {n - 1 \choose r}]

The Binomial Theorem

[(x + y)^n = \sum\limits_{k = 0}^n{ {n \choose k} x^k y^{n - k}]

Other resources

• Wikipedia

Multinomial Coefficient

• Let
  • [n_1 + n_2 + \cdots + n_r = n]
• Then
  • [{n \choose n_1, n_2, \dots, n_r} = \frac{ n!}{ n_1!n_2! \cdots n_r!}]
    • The number of possible divisions of [n] distinct objects into [r] distinct groups of respective sizes [n_1, n_2 \dots, n_r].