高中数学排列组合公式排列组合计算公式

排列组合公式/排列组合 排列 P------和顺序有关 组合 C -------不牵涉到顺序的问题 排列分顺序,组合不分 例如 把5本不同的分给3个人,有几种分法. "排列" 把5本书分给3个人,有几种分法 "组合" 1(排列及计算公式 从n个不同元素中，任取m(m?n)个元素按照一定的顺序排成一列，叫做从n个不同元素中取出m个元素的一个排列;从n个不同元素中取出m(m?n)个元素的所有排列的个数，叫做从n个不同元素中取出m个元素的排列数，用符号 p(n,m)示. p(n,m)=n(n-1)(n-2)„„(n-m+1)= n!/(n-m)!(规定0!=1). 2(组合及计算公式 从n个不同元素中，任取m(m?n)个元素并成一组，叫做从n个不同元素中取出m个元素的一个组合;从n个不同元素中取出m(m?n)个元素的所有组合的个数， 用符号 叫做从n个不同元素中取出m个元素的组合数. c(n,m) 表示. c(n,m)=p(n,m)/m!=n!/((n-m)!*m!);c(n,m)=c(n,n-m); 3(其他排列与组合公式 从n个元素中取出r个元素的循环排列数,p(n,r)/r=n!/r(n-r)!. n个元素被分成k类，每类的个数分别是n1,n2,...nk这n个元素的全排列数为 n!/(n1!*n2!*...*nk!). k类元素,每类的个数无限,从中取出m个元素的组合数为c(m+k-1,m). 排列(Pnm(n为下标，m为上标)) Pnm=n×(n-1)....(n-m+1);Pnm=n～/(n-m)～(注:～是阶乘符号);Pnn(两个n分别为上标和下标) =n～;0～=1;Pn1(n为下标1为上标)=n 组合(Cnm(n为下标，m为上标)) Cnm=Pnm/Pmm ;Cnm=n～/m～(n-m)～;Cnn(两个n分别为上标和下标) =1 ;Cn1(n为下标1为上标)=n;Cnm=Cnn-m 2008-07-08 13:30 welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 公式P是指排列，从N个元素取R个进行排列。公式C是指组合，从N个元素取R个，不进行排列。N-元素的总个数 R参与选择的元素个数 ～-阶乘 ，如 9～,9*8*7*6*5*4*3*2*1 从N倒数r个，表达式应该为n*(n-1)*(n-2)..(n-r+1); 因为从n到(n-r+1)个数为n,(n-r+1),r 举例: Q1: 有从1到9共计9个号码球，请问，可以组成多少个三位数, A1: 123和213是两个不同的排列数。即对排列顺序有要求的，既属于“排列P”计算范畴。 上问题中，任何一个号码只能用一次，显然不会出现988,997之类的组合， 我们可以这么看，百位数有9种可能，十位数则应该有9-1种可能，个位数则应该只有9-1-1种可能，最终共有9*8*7个三位数。计算公式,P(3，9),9*8*7,(从9倒数3个的乘积) Q2: 有从1到9共计9个号码球，请问，如果三个一组，代表“三国联盟”，可以组合成多少个“三国联盟”, A2: 213组合和312组合，代表同一个组合，只要有三个号码球在一起即可。即不要求顺序的，属于“组合C”计算范畴。 上问题中，将所有的包括排列数的个数去除掉属于重复的个数即为最终组合数C(3,9)=9*8*7/3*2*1 排列、组合的概念和公式典型例题分析 例1 设有3名学生和4个课外小组((1)每名学生都只参加一个课外小组;(2)每名学生都只参加一个课外小组，而且每个小组至多有一名学生参加(各有多少种不同方法, )由于每名学生都可以参加4个课外小组中的任何一个，而不限制每个课解(1 外小组的人数，因此共有 种不同方法( (2)由于每名学生都只参加一个课外小组，而且每个小组至多有一名学生参加，因此共有 种不同方法( 点评 由于要让3名学生逐个选择课外小组，故两问都用乘法原理进行计算( 例2 排成一行，其中 不排第一， 不排第二， 不排第三， 不排第四的不同排法共有多少种, 解 依题意，符合要求的排法可分为第一个排 、 、 中的某一个，共3类，每一类中不同排法可采用画“树图”的方式逐一排出: ? 符合题意的不同排法共有9种( 点评 按照分“类”的思路，本题应用了加法原理(为把握不同排法的规律，“树图”是一种具有直观形象的有效做法，也是解决计数问题的一种数学模型( 例, 判断下列问题是排列问题还是组合问题,并计算出结果( (1)高三年级学生会有11人:?每两人互通一封信，共通了多少封信,?每两人互握了一次手，共握了多少次手, (2)高二年级数学课外小组共10人:?从中选一名正组长和一名副组长，共有多少种不同的选法,?从中选2名参加省数学竞赛，有多少种不同的选法, (3)有2，3，5，7，11，13，17，19八个质数:?从中任取两个数求它们的商可以有多少种不同的商,?从中任取两个求它的积，可以得到多少个不同的积, (4)有8盆花:?从中选出2盆分别给甲乙两人每人一盆，有多少种不同welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 的选法,?从中选出2盆放在教室有多少种不同的选法, 分析 (1)?由于每人互通一封信，甲给乙的信与乙给甲的信是不同的两封信，所以与顺序有关是排列;?由于每两人互握一次手，甲与乙握手，乙与甲握手是同一次握手，与顺序无关，所以是组合问题(其他类似分析( (1)?是排列问题，共用了 封信;?是组合问题，共需握手 (次)( (2)?是排列问题，共有 (种)不同的选法;?是组合问题，共有 种不同的选法( (3)?是排列问题，共有 种不同的商;?是组合问题，共有 种不同的积( (4)?是排列问题，共有 种不同的选法;?是组合问题，共有 种不同的选法( 例, 证明 ( 证明 左式 右式( ? 等式成立( 点评 这是一个排列数等式的证明问题，选用阶乘之商的形式，并利用阶乘的性质 ，可使变形过程得以简化( 例5 化简 ( 解法一 原式 解法二 原式 点评 解法一选用了组合数公式的阶乘形式，并利用阶乘的性质;解法二选用了组合数的两个性质，都使变形过程得以简化( 例6 解方程:(1) ;(2) ( 解 (1)原方程 解得 ( (2)原方程可变为 ? ， ， ? 原方程可化为 ( 即 ，解得 第六章 排列组合、二项式定理 一、考纲要求 1.掌握加法原理及乘法原理，并能用这两个原理分析解决一些简单的问题. 2.理解排列、组合的意义，掌握排列数、组合数的计算公式和组合数的性质，并能用它们解决一些简单的问题. 3.掌握二项式定理和二项式系数的性质，并能用它们计算和论证一些简单问题. 二、知识结构 三、知识点、能力点 (一)加法原理乘法原理 说明 加法原理、乘法原理是学习排列组合的基础，掌握此两原理为处理排 列、组合中有关问题提供了理论根据. 例1 5位高中毕业生，准备报考3所高等院校，每人报且只报一所，不同的报名方法共有多少种? welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 解: 5个学生中每人都可以在3所高等院校中任选一所报名，因而每个学生都有3种不同的 报名方法，根据乘法原理，得到不同报名方法总共有 3×3×3×3×3=35(种) (二)排列、排列数公式 说明 排列、排列数公式及解排列的应用题，在中学代数中较为独特，它研 究的对象以及研 究问题的方法都和前面掌握的知识不同，内容抽象，解题方法比较灵活，历届高考主要考查排列的应用题，都是选择题或填空题考查. 例2 由数字1、2、3、4、5组成没有重复数字的五位数，其中小于50 000的 偶数共有( ) A.60个 B.48个 C.36个 D.24个 解 因为要求是偶数，个位数只能是2或4的排法有P12;小于50 000的五位数，万位只能是1、3或2、4中剩下的一个的排法有P13;在首末两位数排定后，中间3个位数的排法有P33，得P13P33P12,36(个) 由此可知此题应选C. 例3 将数字1、2、3、4填入标号为1、2、3、4的四个方格里，每格填一个数字，则每个方格的标号与所填的数字均不同的填法有多少种? 解: 将数字1填入第2方格，则每个方格的标号与所填的数字均不相同的填法有3种，即214 3，3142，4123;同样将数字1填入第3方格，也对应着3种填法;将数字1填入第4方格，也对应3种填法，因此共有填法为 3P13=9(种). 例四 例五可能有问题，等思考 三)组合、组合数公式、组合数的两个性质 说明 历届高考均有这方面的题目出现，主要考查排列组合的应用题，且基本上都是由选择题或填空题考查. 例4 从4台甲型和5台乙型电视机中任意取出3台，其中至少有甲型与乙型电视机各1台，则不同的取法共有( ) A.140种 B.84种 C.70种 D.35种 解: 抽出的3台电视机中甲型1台乙型2台的取法有C14?C25种;甲型2台乙型1台的取法有C24?C15种 根据加法原理可得总的取法有 C24?C25+C24?C15=40+30=70(种 ) 可知此题应选C. 例5 甲、乙、丙、丁四个公司承包8项，甲公司承包3项，乙公司承包1 项，丙、丁公司各承包2项，问共有多少种承包方式? 解: 甲公司从8项工程中选出3项工程的方式 C38种; 乙公司从甲公司挑选后余下的5项工程中选出1项工程的方式有C15种; 丙公司从甲乙两公司挑选后余下的4项工程中选出2项工程的方式有C24种; 丁公司从甲、乙、丙三个公司挑选后余下的2项工程中选出2项工程的方式有C22种. 根据乘法原理可得承包方式的种数有C3 8×C15×C24×C22= ×1=1680(种). (四)二项式定理、二项展开式的性质 说明 二项式定理揭示了二项式的正整数次幂的展开法则，在数学中它是常用的基础知识 ，从1985年至1998年历届高考均有这方面的题目出现，主要考查二welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 项展开式中通项公式等，题型主要为选择题或填空题. 例6 在(x- )10的展开式中，x6的系数是( ) A.-27C610 B.27C410 C.-9C610 D.9C410 解 设(x- )10的展开式中第γ+1项含x6， 因Tγ+1=Cγ10x10-γ(- )γ，10-γ=6,γ=4 于是展开式中第5项含x 6，第5项系数是C410(- )4=9C410 故此题应选D. 例7 (x-1)-(x-1)2，(x-1)3-(x-1)+(x-1),的展开式中的x,的系数等于 解:此题可视为首项为x-1，公比为-(x-1)的等比数列的前5项的和，则其和为 在(x-1)6中含x3的项是C36x3(-1)3=-20x3，因此展开式中x2的系数是-2 0. (五)综合例题赏析 例8 若(2x+ )4=a0+a1x+a2x 2+a3x3+a4x4，则(a0+a2+a4)2-(a1+a3)2的值为( ) A.1 B.-1 C.0 D.2 解:A. 例9 2名医生和4名护士被分配到2所学校为学生体检，每校分配1名医生和2 名护士，不同的分配方法共有( ) A.6种 B.12种 C.18种 D.24种 解 分医生的方法有P22,2种，分护士方法有C24=6种，所以共有6×2,12种不同的分配方法。 应选B. 例10 从4台甲型和5台乙型电视机中任意取出3台，其 中至少要有甲型与乙型电视机各1台，则不同取法共有( ). A.140种 B.84种 C.70种 D.35种 解:取出的3台电视机中，甲型电视机分为恰有一台和恰有二台两种情形. ?C24?+C25?C14=5×6+10×4=70. ?应选C. 例11 某小组共有10名学生，其中女生3名，现选举2 名代表，至少有1名女生当选的不同选法有( ) A.27种 B.48种 C.21种 D.24种 解:分恰有1名女生和恰有2名女生代表两类: ?C13?C1 7+C23=3×7+3=24， ?应选D. 例12 由数学0，1，2，3，4，5组成没有重复数字的 六位数，其中个位数字小于十位数字的共有( ). A.210个 B.300个 C.464个 D.600个 解:先考虑可组成无限制条件的六位数有多少个?应有P15?P 55=600个. 由对称性，个位数小于十位数的六位数和个位数大于十位数的六位数各占一半. ?有 ×600=300个符合题设的六位数. 应选B. 例13 以一个正方体的顶点为顶点的 四面体共有( ). A.70个 B.64个 C.58个 D.52个 解:如图，正方体有8个顶点，任取4个的组合数为C48=70个. welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 其中共面四点分3类:构成侧面的有6组;构成垂直底面的对角面的有2组;形如(ADB1C1 )的有4组. ?能形成四面体的有70-6-2-4=58(组) 应选C. 例14 如果把两条异面直线看成“一对”，那么六棱 锥的棱所在的12条直线中，异面直线共有( ). A.12对 B.24对 C.36对 D.48对 解:设正六棱锥为O—ABCDEF. 任取一侧棱OA(C16)则OA与BC、CD、DE、EF均形成异面直线对. ?共有C16×4=24对异面直线. 应选B. 例15 正六边形的中心和顶点共7个点，以其中三个点 为顶点的三角形共 个(以数字作答). 解:7点中任取3个则有C37=35组. 其中三点共线的有3组(正六边形有3条直径). ?三角形个数为35-3=32个. 例16 设含有10个元素的集合的全部子集数为S，其中由3个元素组成的子集数为T，则 的值为 。 解 10个元素的集合的全部子集数有: S,C010+C110+C210+C310+C410+C510+C610+C710+C810+C910+C1010=2 10=1024 其中，含3个元素的子集数有T=C310=120 故 = 中有4件是次品，从中任意抽了5件 ，至少有3例17 例17 在50件产品 n 件是次品的抽法共 种(用数字作答). 解:“至少3件次品”即“有3件次品”或“有4件次品”. ?C34?C246+C44?C146=4186(种) 例18 有甲、乙、丙三项任务，甲需2人承担，乙、 丙各需1人承担，从10人中选派4人承担这三项任务，不同的选法共有( ). A.1260种 B.2025种 C.2520种 D.5040种 解:先从10人中选2个承担任务甲(C210) 再从剩余8人中选1人承担任务乙(C1 8) 又从剩余7人中选1人承担任务乙(C1 7) ?有C210?C1 8C1 7=2520(种). 应选C. 例19 集合,1，2，3,子集总共有( ). A.7个 B.8个 C.6个 D.5个 解 三个元素的集合的子集中，不含任何元素的子集有一个，由一个元素组成的子集数 C13，由二个元素组成的子集数C23。 由3个元素组成的子集数C33。由加法原理可得集合子集的总个数是 C13+C23+C33+1=3+3+1+1,8 welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury 故此题应选B. 例20 假设在200件产品中有3件是次品，现在从中任意抽取5件，其中至少有两件次品的抽法有( ). A.C23C3197种 B.C23C3197 +C33C2197 C.C5200-C5197 D.C5200-C 13C4197 解:5件中恰有二件为次品的抽法为C23C3197， 5件中恰三件为次品的抽法为C33C2197， ?至少有两件次品的抽法为C23C3197+C33C2197. 应选B. 例21 两排座位，第一排有3个座位，第二排有5个座位，若8名学生入座(每人一个座位)，则不同座法的总数是( ). A.C58C38 B.P12C58C38 C.P58P38 welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury accident occurred, the direct punishment 0.5-10,000 yuan, responsible for the direct responsibility for the accident and the responsibility of leadership, such as concerning administrative sanctions should be dealt with by the personnel Department of the company. (3) personal injury