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排列组合概率题解题技巧

发布者: 老罗 | 发布时间: 2019-4-11 15:59| 查看数: 973| 评论数: 16|帖子模式

1.排列、组合、概率与错位公式
2.排列组合概率解题思路——分类法
3.例题1:繁琐的计算导致正确率变低
4.例题2:通过选项思考暴力破解的可能性
5.例题3:极为简单,一半做错的题
6.例题4:分不同情况考虑安排方案
7.例题5:分不同情况考虑安排方案
8.例题6:理解排列组合题的关键
一、排列、组合、概率与错位公式
「数量关系」板块中的「排列、组合、概率」方面的题目每年必考、国考省考都会考,而此类题的难度一般较高,因此掌握它们的解题方法是非常有必要的。
总体来说,此类题目的公式非常简单,大致只有三个半,即排列公式、组合公式、概率公式和错位排列公式。
(1)排列公式
A(总个数,选出排列的个数)
特点是每个个体有「排列」的独特性,谁先选、谁后选会影响结果。
例如5个人选3个排队,5个项目选3个先后完成,两种情况的运算均为:
A(5,3)=5×4×3=60种方式
(2)组合公式
C(总个数,选出组合的个数)
特点是每个个体没有「排列」的独特性,谁先选、谁后选都不影响结果。
例如5个人选3个参加比赛,5个项目选3个于今年内完成(不要求完成顺序),则运算均为:
C(5,3)=C(5,2)
=5×4÷(1×2)=10种方式
注意C(5,3)一般要转换为C(5,2),其原因是:
C(5,3)=5×4×3÷(1×2×3)=5×4÷2,中间要约去3,因此可能会多花两三秒钟,故要尽量节约时间。
注:排列组合公式很好记忆,由于公考中考察的「排列组合概率」题的数值不会很大,因此在实际考试中,直接在纸上用笔列草稿即可:
总数×(总数-1)×(总数-2)×……
一直让相乘数字的个数达到「选出的个数」,即为排列公式;
再从1开始乘,乘到「选出的个数」,用排列公式得出的结果除以该数即为「组合公式」。
关于「排列组合」,最标准的公式如下:
这两个公式很优美,不过大家实际做题时没必要这么列,毕竟公考中的n和m都不会很大,一边列公式一边约分(尤其是对于组合公式)即可。
只要熟练掌握「排列组合」公式,理解两者的不同,就很容易解出答案。
(3)概率公式
发生某情况的概率=发生该情况的个数/总情况的个数
概率公式极为简单,也很好理解,而「总情况个数」一般也能快速得出,此类题的解题关键是「发生该情况的个数」。
(4)错位排列公式
此类公式只能算「半个公式」,因为它基于排列组合公式,但公式的步骤又很难理解,而且它虽然在公考中出现过,但出现次数极少,因此大家只要记住它的描述和数值即可。
错位排列的描述为「全部错位」,例如:
一个人写了n封不同的信及相应的n个不同的信封,他把这n封信都装错了信封,问都装错信封的装法有多少种?
上面这道题就是「错位排列」的最初源头,类似描述包括「5个部门5个人员重新分配,都不回到原部门」等。
「错位排列」的数据很好记忆,总共只有3个(用D表示):
D1= 0,D2= 1,D3=2,D4= 9,
D5= 44,D6= 265,D7= 1854。
D1、D2太小,D7及以上太大,一般不会考;D3可直接从纸上列出情况,很好理解。只要记住D4~D6的结果即可。
二、排列组合概率解题思路——分类法
根据上面的描述可发现,「排列组合」题的公式一点都不难,而且也很好记忆。此类题的难点主要在于「确定其属于什么类别」。
在实际考试中,「排列」「组合」「概率」三者经常结合在一起,往往一道求概率的题,其分情况和总情况都需要用「排列组合公式」去求得结果。
根据公考出现的题目,可将其大致分为以下几类(有时候下面几类会再次结合):
(1)加法类
求某事物的概率,该事物有多种情况成立,则总概率等于每种情况成立时的概率相加。
求某情况的总数,该情况分为多种分情况,则总情况等于所有情况的和。
(2)乘法类
此类题目的描述和加法类有所类似,区别的关键在于某概率成立/某情况成立时和分概率/分情况的关系。
求某事物的概率,该事物分为多种情况,当所有情况成立时才满足题干要求,则总概率等于每种情况成立时的概率相乘。
求某情况的总数,该情况为多种分情况的总体组合,每种分情况都有自己的个数,则总情况等于所有分情况相乘。
用一个简单例题来区别「加法类」和「乘法类」的区别:
甲乙下棋(没有平局),甲每盘战胜乙的几率为40%,三局两胜,求甲三局后战胜乙的几率。
此时可将其分为「甲3胜」和「甲2胜1负」两种情况,然后将两种情况相加即可,即:
(40%×40%×40%)+C(3,1)×(40%×40%×60%)
甲乙下棋(没有平局),甲每盘战胜乙的几率为40%,三局两胜,求甲通过「先输一局、再赢两局」这种方法战胜乙的几率。
此时每盘情况都固定,则结果为:
60%×40%×40%
此类题在没有概率的「排列组合」题中也存在。例如甲乙两个部门选3人参加活动:
如果要求是「分情况」,例如共有「甲1乙2」「甲2乙1」「甲3乙0」3种情况,则需要分不同情况得出结果后相加。
如果要求是「分部门」,例如「甲1乙2」的形式固定下来了,则总情况即为「甲1」的情况数×「乙2」的情况数。
很多「排列组合概率」的难题可能同时出现两种情况,只要能将其分类分清楚了,其实这种题目并不难。
(3)特殊类(除错位排列)
某些难题可能会考察特殊情况的排列组合,例如:
「植树时在马路两侧植树且第一棵树固定」
「2人一组,共有多组参加活动」
「在圆桌上参加宴会」
「有的人可选择任何位置,有的人只能选择部分位置(如住旅馆只能住在1层等)」
这些情况本质上和「排列组合」公式以及「加法、乘法」的分类是想通的,除了「错位排列」之外,其他题目都是非常好理解的,只要根据题干描述进行分类即可,在接下来的真题讲解中都会详细分析。
需要注意,如果题目看似是在求「排列组合概率」,但选项和题干数字都很小,那很可能需要使用「逐个列出」等方法去解题。关于这方面的解析,各位小伙伴可参考之前的内容:「数量关系」解题技巧(7)——整消法。
三、例题1:繁琐的计算导致正确率变低
【2017国考地市级卷66题/ 省级卷68题】小张需要在5个长度分别为15秒、53秒、22秒、47秒、23秒的视频片段中选取若干个,合成为一个长度在80~90秒之间的宣传视频。要求每个片段均需完整使用且最多使用一次,并且片段间没有空闲时段。
小张最多可能做出多少个不同的视频?
(A)6
(B)12
(C)18
(D)24
正确答案C,正确率50%,易错项B
列出题干数据关系:
①5片段长度为15、53、22、47、23
②合成视频长度80~90
③片段完整、无空闲、最多使用一次,求视频种类数量
由①②可知,小张需要选择几个视频片段,找出时间相加在80~90之间的组合。
把几个数从大到小排列:53、47、23、22、15,首先从最大数53开始罗列所有的可能:
53+47=100>90,排除
53+23=76,76+(最小的)15=91>90,排除
53+22+15=90,符合情况
然后从47开始数:
47+23=70,70+22=92>90,排除
47+23+15=85,符合情况
47+22+15=84,符合情况
可以看出,符合情况的共三类,分别为:
53+22+15=90
47+23+15=85
47+22+15=84
根据③可知,每个视频片段放在不同的位置都是不同的视频,即本题适用排列公式(A),不适用组合公式(C),可得视频数为:
A(3,3)+A(3,3)+A(3,3)
=6+6+6=18个,C选项正确。
此类计算量大的题目一定要有耐心才能解得正确答案,需要注意本题适用于排列公式。
虽然这道题的计算量不是很大,但计算较为繁琐,因此正确率不高。
四、例题2:通过选项思考暴力破解的可能性
【2017国考省级卷70题】某集团企业5个分公司分别派出1人去集团总部参加培训,培训后再将5人随机分配到这5个分公司,每个分公司只分配1人。
5个参加培训的人中,有且仅有1人在培训后返回原分公司的概率为:
(A)低于20%
(B)在20%~30%之间
(C)在30%~35%之间
(D)大于35%
正确答案D,正确率15%,易错项B
列出题干数据关系:
①5公司分别派1人
②重新分配,每公司分配1人
③求有且仅有1人返回原公司的概率
列出计算公式:
有且仅有1人返回原公司的概率=有且仅有1人返回原公司的情况/全部分配情况
根据②可知,5个人分到不同的公司属于不同的分配情况,符合排列公式(A),即:
全部分配情况=A(5,5)=120
本题的难点是「只有1人返回原公司的分配情况」。设5家公司为ABCDE,5名员工也为ABCDE,字母一一对应。以员工A为例,该描述可以分解为两句话:
(1)员工A返回了A公司;
(2)其他4名员工没有回到自己的公司,即B可以去CDE不能去B,C可以去BDE不能去C……
分析之后可得出,(2)是个典型的4个元素的错位排列问题,即D4=9。
错位排列公式:D3=2,D4=9,D5=44,D6=265,更复杂的一般不会去考察。
BCDE员工返回原公司的概率和A员工相同,共有9×5=45种分配情况。因此,所求概率为:
45/120=37.5%>35%,D选项正确。
那么问题就来了:如果考生不熟悉错位排列的公式,或者不熟悉错位排列的适用场景,应该怎么办呢?
这就是国考的精髓之处。相对于排列组合公式,错位排列是一个较为冷门的考点,但本题并不要求考生一定要掌握,其解题奥秘,就在原文中。
通过分析我们不难看出,全部的分配情况为A(5,5)=120,而ABCDE公司的ABCDE员工没有特殊要求,因此:
120=5×「员工A返回A公司,其他4名员工没有回到自己的公司」的分配情况(即员工A返回A公司这一情况没有特殊性,BCDE公司和员工也符合)
可知「员工A返回A公司,其他4名员工没有回到自己的公司」的分配情况=24
观察选项可知,本题数值最大选项D也只有35%,而24的35%约比8大一点(35%比33.33%大一点,24×33.33%=8),即:
「最多只需要数出9种情况就能得到正确答案」
也就是说,本题可以暴力破解,一个个数所有的分配可能即可,不会浪费太多时间。
那么,以上文说的那个情况为例:A员工返回了A公司,其他4名员工没有回到自己的公司,即B可以去CDE不能去B,C可以去BDE不能去C……
在这种情况下,以员工B去C公司为例,C只能去BDE。如果C去B,那么D只能去E,E只能去D;如果C去D,那么D只能去E,E只能去B;如果C去E,那么D只能去B,E只能去D。也就是说,B去C的前提下,只有3种情形。同样,B去D、E也是各有3种情形,也就是共有9种。
以B去C,C去B为例简单列图就能明白这个关系了(红箭头代表B去C,蓝箭头代表其他所有可能)。
可能性一:
75747b81ea4d14bd28e5f0c057f328f1.jpg
可能性二:
d6ee65acdec97ebaad5487cd77beb894.jpg
可能性三:
7e1e007025a8083ae5bbe571233ffb8f.jpg
虽然上述内容文字描述看上去很复杂,但在草稿纸上列表就是半分钟的事情。这种解法也可得出正确答案。
之所以把这个「不知道、不会用错位排列」的解题方法写了这么多,是因为要给各位小伙伴提供另一种一个思考角度,通过选项思考暴力破解的可能性。本题正确率只有15%,如果做对就战胜了绝大多数考生,因此千万不要轻言放弃。
五、例题3:极为简单,一半做错的题
【2015国考地市级卷67题/省级卷66题】把12棵同样的松树和6棵同样的柏树种植在道路两侧,每侧种植9棵,要求每侧的柏树数量相等且不相邻,且道路起点和终点处两侧种植的都必须是松树。
共有多少种不同的种植方法?
(A)36
(B)50
(C)100
(D)400
正确答案C,正确率51%,易错项B
列出题干数据关系:
①12松6柏种两侧,每侧9棵
②柏每侧相等(各3棵),不相邻
③起点终点都是松
根据①②可知每侧固定6松3柏
根据③可知每侧两端的树固定为松
实际情况如下:
fdfae503b4cc38a2e2b33ea833d292d9.jpg
两端加粗的「松」有固定要求,6松内部共有5个可以插入的空(即满足「柏不相邻」的要求)。
也就是说,本题可以理解为「从5个可以插入的空中,选出3个空种植柏」。由于本题的柏没有特征,符合组合公式,因此每侧种植方法为:C(5,3)=10
两侧总共种植方法为10²=100,C选项正确。
在本题中,「两侧种植情况相同」这个情况能帮助考生秒排除B,如果答案中有更多的非平方数,例如30、50、100、120,那么可以立即选出100。
「不相邻」是排列组合题中非常流行的考法,一定要引起注意。
六、例题4:看似简单叙述中的隐藏陷阱
【2015国考地市级卷68题/省级卷67题】某单位有3项业务要招标,共有5家公司前来投标,且每家公司都对3项业务发出了投标申请,最终发现每项业务都有且只有1家公司中标。
如5家公司在各项业务中中标的概率均相等,这3项业务由同一家公司中标的概率为多少?
(A)1/25
(B)1/81
(C)1/125
(D)1/243
正确答案A,正确率21%,易错项C
列出题干数据关系:
①3项业务,5家公司投标
②每项业务1家公司中标
③求同一家公司中标的概率
根据①②可知,某家公司某项业务中标几率为:
1÷5=1/5
共有3项业务,则某家公司3项业务全部中标几率为:
(1/5)³=1/125
题干说的是「同一家公司」,并没有说是「(固定的)某家公司」,因此「同一家公司3项业务全部中标几率」为:
1/125×5=1/25,A选项正确。
本题基本没有难度,但错误率极高。很多考生不是不会做,而是没有认真审题,没有理解「同一家公司」的含义。这道题乍一眼看上去很像送分题,概率的计算公式非常简单,数值也很小,看似平平淡淡,但考场上并不会标注本题的正确率。如果事先把正确率告诉考生,很多考生就能意识到叙述中暗含的陷阱了。
从这道题可以看出,「审题」非常重要,看上去很简单的叙述也可能有陷阱。
七、例题5:分不同情况考虑安排方案
【2014国考71题】一次会议某单位邀请了10名专家。该单位预定了10个房间,其中一层5间。二层5间。已知邀请专家中4人要求住二层、3人要求住一层。其余3人住任一层均可。那么要满足他们的住宿要求且每人1间。
有多少种不同的安排方案?
(A)75
(B)450
(C)7200
(D)43200
正确答案D,正确率46%,易错项C
列出题干数据关系:
①10人住10房间,每人一间
②一层5间二层5间
③4人二层,3人一层,3人任意层
④求安排方案的数量
根据③的限定可逐层考虑安排情况,并将不同的情况相乘即可。
二层4人住5间,符合排列公式,即:
A(5,4)=5×4×3×2=120
二层3人住5间,符合排列公式,即:
A(5,3)=5×4×3=60
还有3人住余下3间,符合排列公式,即:
A(3,3)=3×2=6
因此总安排情况=三种情况相乘
=120×60×6
=7200×6
=43200种,D选项正确。
本题一定要注意「3人任意层」的含义是「安排好一层、二层人员之后,还余下3间房,3人在3间房中任意挑选」,而不是「3人住3间只有一种情况」。如果没有理解这一点,就很容易误选C。
一定要准确理解题干描述,不要在简单题目上丢分。
八、例题6:理解排列组合题的关键
【2012国考70题】有5对夫妻参加一场婚礼,他们被安排在一张10个座位的圆桌就餐,但是操办者不知道他们之间的关系,随机安排座位。
5对夫妻恰好相邻而坐的概率是多少?
(A)≤1‰
(B)1‰~5‰
(C)5‰~1%
(D)>1%
正确答案A,正确率31%,易错项B
列出题干数据关系:
①5对夫妻,一个圆桌
②10个座位,随机安排
③恰好相邻,求其概率
③所要求的概率为:
5队夫妻恰好相邻的安排数量/总安排数量
需要注意本题是「一个圆桌」,即夫妻ABCDE和BCDEA、CDEAB、DEABC、EABCD的排列情况是相同的,也就是说,根据①将5队夫妻视为整体,则整体安排数量为:
A(5,5)÷5=2×3×4
夫妻内部有夫左妻右、夫右妻左两种情况,因此5队夫妻内部的排列情况为2的5次方,即5队夫妻恰好相邻的安排数量为:
2×3×4×2的5次方
10人同样位于「一个圆桌」,同理其总安排数量为:
A(10,10)÷10=2×3×……×9
即:
5队夫妻恰好相邻的安排数量/总安排数量
=2×3×4×2的5次方/(2×3×……×9)
=2的5次方/5×6×7×8×9
=2/5×3×7×9=2/945,A选项正确。
本题即使不考虑「圆桌」的排列,最后结果也是1/945,同样位于A选项范围内。总体来看,这道题还是很人性化的。
理解了本题,就理解了大部分公考中的「排列组合」题。

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