To derive C(n,r) by the product rule
Task 1: select one element from a set of n elements
Task 2: select the remaining (r – 1) elements from the set (with n – 1 elements
remaining)
Therefore, C(n,r) = n C(n – 1,r – 1).
Are the above combinatorial argument and the formula correct? Please give reasons.
(b) Use the combinatorial argument to show that
n C(n – 1,r) = (r + 1) C(n,r + 1)
Hints: consider the selection of a leader and r team members from n people.
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Anyone understand it ?i have no ideas on it.
Task 1: select one element from a set of n elements
任務一：從n個元素中選出1個
Task 2: select the remaining (r – 1) elements from the set (with n – 1 elements
remaining)
任務二：從剩下的(n - 1)個元素中選出(r - 1)個
Therefore, C(n,r) = n C(n – 1,r – 1).
因此，(從n個中選出r個的方法數) = (從n個元素中選出1個的方法數)*(從剩下的(n - 1)個元素中選出(r - 1)個的方法數)
即 C(n,r) = n C(n – 1,r – 1)
Are the above combinatorial argument and the formula correct? Please give reasons.
上述的推論以及公式正確嗎？請說明原因。
(b) Use the combinatorial argument to show that
n C(n – 1,r) = (r + 1) C(n,r + 1)
利用組合的推論方式證明 n C(n – 1,r) = (r + 1) C(n,r + 1)
Hints: consider the selection of a leader and r team members from n people.
提示：考慮從n個人中選出1個隊長和r個隊員的方法數。
[答]
(a) 推論及公式均錯誤。
(b)
計算從n個人中選出1個隊長和r個隊員的方法數時，有兩種方法。
(i) 先選隊長再選隊員：C(n,1)*C(n-1,r) = n C(n – 1,r)
(ii) 先選出r+1個人再從中選出隊長：C(n,r+1)*C(r+1,1) = (r + 1) C(n,r + 1)
(i)(ii)算的是同一件事的方法數，∴ n C(n – 1,r) = (r + 1) C(n,r + 1)
得證。