June 8, 2021 1 min to read

Snow White and the Seven Dwarfs - Combination

Nàng Bạch Tuyết và Bảy chú lùn - Toán tổ hợp

• Combinatorics notations

  1. \(\binom{n}{k}\) = The number of ways to choose k elements out of n elements. In some books, you might also see \(C_{n}^k\).

  2. \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\)

• Fomulas

  The number of ways to order n elements on a row = \(n!\).

  Similarly, we have \(n!\) ways to arrange elements on a circle, but because the circle can be rotated \(n\) times, the number of ways to order n elements on a circle is \[\frac{n!}{n} = (n-1)! \] The number of ways to select k out of n elements is \[\binom{n}{k}=\frac{n!}{k!(n-k)!}\] This is the \(\textbf{combination}\) of k elements, they don't bother order of selection. But \(\textbf{permutation}\) does. The permutation of k out of n elements is \[P_{n}^k=\frac{n!}{(n-k)!}\]Make sure not to confuse between the two notations.

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