1 . (1)已知
、
为正整数,
,求证:
:
(2)已知
、
为正整数,求证:
;
(3)
、
为正整数,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b4b3879d1c6debf0333008f686634e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e10e0bb04d7d261d880aea655e19db1.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0617f82b34b84b66f789c2dc3a397ae6.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d298057595d012ad2665c5433a9cc43.png)
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名校
2 . 富比尼原理又称算两次原理,是组合数学中非常重要的计算方法,下面的组合恒等式可以用富比尼原理进行证明,具体如下:
人中有1人是军人,从
人中选
人各奖励1颗星,共有
种选法,另一方面,这等价于考虑这
人中的军人是否被选中,若选中军人,则有
种选法,若未选中军人,则有
种选法,所以
;
(1)若
,求关于
的方程
的解;
(2)将题干中的问题推广到
人中有
人是军人的情形,写出结论并加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2c95df1e638ebfe52ab35c93d4e424c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7707aba7b9fed2c8e4704b82ce09087a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b1dac9ce7c08de49a30f1d982bbedee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894e4655d43814eae5e21ca44840d918.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4746217439e9cbcfe130bc074736e86b.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b54fdc920fd0a627959bbc5bd292d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/875978e3eeb54751d2ce83ae18421657.png)
(2)将题干中的问题推广到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b367cda0011d8968cb46206aa0bb4aa.png)
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3 . 由
,
,
,
,
,
,
,
,
,
按任意顺序组成的没有重复数字的数组,记为
,设
,其中
.
(1)若
,求
的值;
(2)求证:
;
(3)求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b06e95b57b7a81cd81d05557a11fa92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3304e23f3b0f9569c4140ca89b6498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d02ea8c4988c5c28ab93f0d70fb55a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d41baa5854760c61b58e4813c0dc23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da3b9d0ea6fb9bfc3f298783bb156934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a926a5378a8dc9f9097eeb8ad3ac1d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202957f2a77c9db7845629c5a4cd8c05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3293f12a66981b2d973b5ea27c319ed6.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e873ae3fcc2ceb24991d525073b7e8b.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3293f12a66981b2d973b5ea27c319ed6.png)
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