MT【137】多少个?

数列\(\{a_n\}\)共11项,\(a_1=0,a_{11}=4\),且\(|a_{k+1}-a_{k}|=2,k=1,2,\cdots,10\)
求满足条件的不同的数列的个数______

解答:\(a_{k+1}-a_{k}\)为2或者-2.设有 \(x\)个2,和\(10-x\)个-2.故\(2x-2(10-x)=4\),得\(x=6\)故有\(C_{10}^6\)不同的数列.

练习:
数列\(\{a_n\}\)共21项,满足\(|a_{k+1}-a_{k}|=1,(k=1,2,\cdots,20)\)若\(a_1,a_7,a_{21}\)成等比数列,且\(a_1=0,a_{21}=9\),
求满足条件的不同的数列的个数______
答案:15099,提示：按上面方法,\(a_7\)分\(3\)或者\(-3\)两类讨论.