概率论

         C n m C_{n}^{m} Cnm​ 的意义: 从n个元素中, 取m个元素的不同组合个数.          如: 从 A, B, C 三个字母中, 取两个字母的不同组合个数?          A, B, C 的两两组合为: AB, BC, AC.          n = 总字母个数, m = 取几个字母组合,          则: 表示为: C 3 2 C_{3}^{2} C32​

        公式: C n m = n ! m ! ⋅ ( n − m ) ! C_{n}^{m} = \frac{n!}{m!\cdot (n-m)!} Cnm​=m!⋅(n−m)!n!​ n ! = n ∗ ( n − 1 ) ∗ ( n − 2 ) ∗ ( n − 3 ) ∗ ⋯ ⋯ ∗ 1 n! =n*(n-1) *(n-2)*(n-3)*\cdots\cdots*1 n!=n∗(n−1)∗(n−2)∗(n−3)∗⋯⋯∗1 如: n = 5 n=5 n=5 则: 5 ! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 5!=5*4*3*2*1 5!=5∗4∗3∗2∗1

如: n = 7 n=7 n=7 则: 7 ! = 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 7!=7*6*5*4*3*2*1 7!=7∗6∗5∗4∗3∗2∗1

特殊情况: 1 ! = 1 1!=1 1!=1 , 0 ! = 1 0!=1 0!=1

计算(1):          C 3 2 = ? C_{3}^{2} = ? C32​=?,          n = 3 , m = 2 n=3, m=2 n=3,m=2 带入公式: n ! m ! ⋅ ( n − m ) ! \frac{n!}{m!\cdot (n-m)!} m!⋅(n−m)!n!​ = 3 ! 2 ! ⋅ ( 3 − 2 ) ! =\frac{3!}{2!\cdot (3-2)!} =2!⋅(3−2)!3!​ = 3 ∗ 2 ∗ 1 2 ∗ 1 ⋅ ( 1 ) ! =\frac{3*2*1}{2*1\cdot (1)!} =2∗1⋅(1)!3∗2∗1​ = 6 2 ∗ 1 =\frac{6}{2 * 1} =2∗16​ = 6 2 =\frac{6}{2} =26​ = 3 =3 =3          C 3 2 = 3 C_{3}^{2} = 3 C32​=3

计算(2):          C 5 3 = ? C_{5}^{3} = ? C53​=?,          n = 5 , m = 3 n=5, m=3 n=5,m=3 带入公式: n ! m ! ⋅ ( n − m ) ! \frac{n!}{m!\cdot (n-m)!} m!⋅(n−m)!n!​ = 5 ! 3 ! ⋅ ( 5 − 3 ) ! =\frac{5!}{3!\cdot (5-3)!} =3!⋅(5−3)!5!​ = 5 ! 3 ! ⋅ ( 2 ) ! =\frac{5!}{3!\cdot (2)!} =3!⋅(2)!5!​ = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 3 ∗ 2 ∗ 1 ⋅ ( 2 ∗ 1 ) ! =\frac{ 5 * 4 * 3 * 2 * 1 }{ 3 * 2 * 1 \cdot (2 * 1) !} =3∗2∗1⋅(2∗1)!5∗4∗3∗2∗1​ = 120 6 ∗ 2 =\frac{120}{ 6 * 2} =6∗2120​ = 120 12 =\frac{120}{12} =12120​ = 10 =10 =10          C 5 3 = 10 C_{5}^{3} = 10 C53​=10