조합(Combination)이란, 서로 다른 n {\displaystyle n} 개 중 순서를 무시하고 r {\displaystyle r} 개를 택하는 것이다. 기호로 n C r {\displaystyle {\scriptstyle n}\mathrm {C} {\scriptstyle r}} 로 나타낸다.
n C r {\displaystyle {\scriptstyle n}\mathrm {C} {\scriptstyle r}} 에서 n ≧ r {\displaystyle {n}\mathrm {\geqq } {r}}
n C r = n P r r ! = n ( n − 1 ) ( n − 2 ) ⋯ ( n − r + 1 ) r ! = n ! r ! ( n − r ) ! ( 0 ≤ r ≤ n ) {\displaystyle {\begin{array}{lll}{n\mathrm {C} r}&{=}&{\mathrm {\frac {nPr}{r!}} }\\{}&{\mathrm {=} }&{\mathrm {\frac {{n}{(}{n}{-}{1}{)(}{n}{-}{2}{)}\cdots {(}{n}{-}{r}{+}{1}{)}}{r!}} }\\{}&{=}&{\mathrm {\frac {n!}{{r}{!(}{n}{-}{r}{)!}}} }\end{array}}{(}{0}\leq {r}\leq {n}{)}}
( 1 ) n C r = n C n − r ( 0 ≤ r ≤ n ) ( 2 ) n C 0 = 1 , n C n = 1 , n C 1 = n ( 3 ) n C r = n − 1 C r + n − 1 C r − 1 ( 1 ≤ r < n ) {\displaystyle {\begin{aligned}&{{(}{1}{)}{\scriptstyle n}{\mathrm {C} }{\scriptstyle r}\mathrm {=} {\scriptstyle n}{\mathrm {C} }{\scriptstyle n}{\scriptstyle \mathrm {-} }{\scriptstyle r}\;{\mathrm {(} }{0}\mathrm {\leq } {r}\mathrm {\leq } {n}{\mathrm {)} }}\\&{{\mathrm {(} }{2}{\mathrm {)} }{\scriptstyle n}{\mathrm {C} }{\scriptstyle 0}\mathrm {=} {1}{\mathrm {,} }\;{\scriptstyle n}{\mathrm {C} }{\scriptstyle n}={1}{,}\;{\scriptstyle n}{\mathrm {C} }{\scriptstyle 1}\mathrm {=} {n}}\\&{{(}{3}{\mathrm {)} }{\scriptstyle n}{\mathrm {C} }{\scriptstyle r}{\scriptstyle =}{\scriptstyle n}{\scriptstyle -}{\scriptstyle 1}{\mathrm {C} }{\scriptstyle r}\mathrm {+} {\scriptstyle n}{\scriptstyle -}{\scriptstyle 1}{\mathrm {C} }{\scriptstyle r}{\scriptstyle \mathrm {-} }{\scriptstyle 1}\;{(}{1}\mathrm {\leq } {r}<{n}{)}}\end{aligned}}}
S o l u t i o n ) 20 C 3 = 20 P 3 3 ! = 20 ⋅ 19 ⋅ 18 3 ⋅ 2 ⋅ 1 = 1140 {\displaystyle {Solution}{\mathrm {)} }\;{\begin{array}{lll}{{\scriptstyle \mathrm {20} }\mathrm {C} {\scriptstyle 3}}&{=}&{\frac {{\scriptstyle 20}\mathrm {P} {\scriptstyle 3}}{3\mathrm {!} }}\\{}&{=}&{\frac {{20}\cdot {19}\cdot {18}}{{3}\cdot {2}\mathrm {\cdot } {1}}}\\{}&{=}&{1140}\end{array}}}
