题目:
(1)计算:C33+C43+C53+…+C103
(2)证明:Ank+kAnk-1=An+1k.
答案:
(1)∵Cmn+Cm-1n=Cmn+1,
∴原式=C44+C43+C53+…+C103
=C54+C53+C63+…+C103
=C64+C63+C73+…+C103
=…
=C104+C103
=C114
=330
(2)证明:∵
=A nm n! (n-m)!
∴左边=
+kn! (n-k)! n! (n-k+1)!
=n![(n-k+1)+k] (n-k+1)!
=(n+1)! (n-k+1)!
=An+1k=右边