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 Thinking Operations Binomial theorem⦿ Binomial theorem Combinatorics⦿ Permutations⦿ Combination⦿ Variation Sets⦿ Operations on Sets Logic⦿ Logical Operations and Truth Tables

Combinatorics

Combination

Combination without Repetition

A combination is a way of selecting k items from a collection of n items (k ≤ n), such that (unlike permutations) the order of selection does not matter. The repetition of items is not allowed.

The number of combinations::

${C}_{n}^{k}=\frac{n·\left(n-1\right)·\left(n-2\right)·...\left(n-k+1\right)}{k!}=\frac{n!}{k!·\left(n-k\right)!}=\left(\genfrac{}{}{0}{}{n}{k}\right)$

Example:

From 5 items {a,b,c,d,e} choose 2,  repetition is not allowed:${C}_{5}^{2}=\frac{5!}{2!·\left(3\right)!}=\left(\genfrac{}{}{0}{}{5}{3}\right)=10$

(a,b), (a,c), (a,d), (a,e), (b,c), (b,d), (b,e), (c,d), (c,e), (d,e)

Combination without Repetition

A combination is a way of selecting k items from a collection of n items, such that (unlike permutations) the order of selection does not matter. The repetition of items is allowed.

The number of combinations:

${\overline{C}}_{n}^{k}=\left(\genfrac{}{}{0}{}{n+k-1}{k}\right)$

Example:

From 4 items {a,b,c,d} choose 2 items, repetition is allowed:

The number of combinations: ${\overline{C}}_{4}^{2}=\left(\genfrac{}{}{0}{}{4+2-1}{2}\right)=\left(\genfrac{}{}{0}{}{5}{2}\right)=10$

(a,a), (a,b), (a,c), (a,d), (b,b), (b,c), (b,d), (c,c), (c,d), (d,d)