C(n,k)=C(n-1,k-1)+C(n-1,k)为什么这个等式成立?请大神帮我解释下C(n,k)=C(n-1,k-1)+C(n-1,k)为什么这个等式成立?请大神帮我解释下下面这段解释,我没理解,谢谢.Whydoesthiswork?Considerwhetherthenthe

Question

C(n,k) = C(n-1,k-1) + C(n-1,k) 为什么这个等式成立?请大神帮我解释下
C(n,k) = C(n-1,k-1) + C(n-1,k)
为什么这个等式成立?请大神帮我解释下下面这段解释,我没理解,谢谢.
Why does this work? Consider whether the nth element appears in one of the C(n,k)
subsets of k elements. If so, we can complete the subset by picking k 1 other items from the other n 1. If not, we must pick all k items from the remaining n 1. There is no overlap between these cases, and all possibilities are included, so the sum counts all k-subsets.

Answer 1

很简单的问题
把n个元素分成两组
第一组n-1个,第二组1个
从中取出k个元素【方法有 C(n,k)种】 ,取法有两种
(1)从第一组中取出k个
方法有C(n-1,k)种
(2) 从第一组中取出k-1个,从第二组中取出1个
方法有C(n-1,k-1)种
所以,
C(n,k)
=C(n-1,k)+C(n-1,k-1)