# 四、公式

abc abc, acb, bac, bca, cab, cba
abd abd, adb, bad, bda, dab, dba
acd acd, adc, cad, cda, dac, dca
bcd bcd, bdc, cbd, cdb, dbc, dcb

==> 组合 = 排列 / 排序方案数

``````C(n, m) = A(n, m) / m!
= n! / [m! (n - m)!]
``````

# 五、计算

### 例1

C(3, 0) = 1
C(3, 1) = A(3, 1) / 1! = 3 / 1! = 3
C(3, 2) = A(3, 2) / 2! = 3 * 2 / 2 = 3
C(3, 3) = A(3, 3) / 3! = 3 ! / 3! = 1

### 例2

C(4, 0) = 1
C(4, 1) = A(4, 1) / 1! = 4
C(4, 2) = A(4, 2) / 2! = 4 * 3 / 2 = 6
C(4, 3) = A(4, 3) / 3! = 4 * 3 * 2 / 3! = 4
C(4, 4) = A(4, 4) / 4! = 4! / 4! = 1

### 例3

C(5, 0) = 1
C(5, 1) = A(5, 1) / 1! = 5
C(5, 2) = A(5, 2) / 2! = 5 * 4 / 2! = 10
C(5, 3) = A(5, 3) / 3! = 5 * 4 * 3 / 3! = 10
C(5, 4) = A(5, 4) / 4! = 5 * 4 * 3 * 2 / 4! = 5
C(5, 5) = A(5, 5) / 5! = 5! / 5! = 1

# 六、两个重要的性质

（一）从上面的三个例题中，可以发现一个规律：

``````C(n, m) = C(n, n - m)
``````

``````  C(n, n - m)
= n! / {(n - m)! [n - (n - m)]!}
= n! / [(n-m)! m!]
= C(n, m)
``````

（二）

``````C(n, m) = C(n - 1, m) + C(n - 1, m - 1)
``````

1 不开除班长，从剩下n - 1个里开除m个
2 开除班长，再从剩下n - 1个里开除m - 1个

combination.png

qrcode_for_kidscode_258.jpg

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