In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:

1) The marksman first chooses a column from which a target is to be broken.

2) The marksman must then break the lowest remaining target in the chosen column.

If the rules are followed, in how many different orders can the eight targets be broken?

By the Multiplication Principle, the answer is：

ways to choose from Left column   ways to choose from Middle column  ways to choose from Right column 

The AIME (American Invitational Mathematics Examination) is an intermediate examination between the AMC 10 or AMC 12 and the USAMO. All students who took the AMC 12 and achieved a score of 100 or more out of a possible 150 or were in the top 5% are invited to take the AIME. All students who took the AMC 10 and had a score of 120 or more out of a possible 150, or were in the top 2.5% also qualify for the AIME. There is no additional registration fee for the American Invitational Mathematics Examination, unless you choose to take the second sitting of the exam. An additional administrative/shipping fee is charged, with a minimum of for the first 10 students and for over 10 students. This is explained more thoroughly in the AMC 10/12 and AIME Teacher's Manuals.

AIME（美国数学邀请考试）是介于 AMC 10 或 AMC 12 和 USAMO 之间的中级考试。所有参加AMC 12并在满分150分中取得100分或以上或排名前5%的学生均可应邀参加AIME考试。所有参加 AMC 10 考试并在 150 分的满分中取得 120 分或以上，或排名前 2.5%的学生也有资格参加 AIME 考试。美国数学邀请考试不收取额外的报名费，除非您选择参加第二次考试。前 10 名学生和超过 10 名学生需支付额外的管理费/运费。AMC 10/12 和 AIME 教师手册对此有更详细的解释。

Content

The AIME is a 15 question, 3 hour examination in which each answer is an integer number from 0 to 999. The questions on the AIME are much more difficult and students are very unlikely to obtain the correct answer by guessing. As with the AMC 10 and AMC 12 (and the USAMO), all problems on the AIME can be solved by pre-calculus methods. The use of calculators is not allowed.

考试内容

AIME 考试共 15 道题，考试时间为 3 小时，每道题的答案都是 0 到 999 之间的整数。AIME 的试题难度更大，学生很难通过猜测获得正确答案。与 AMC 10 和 AMC 12（以及 USAMO）一样，AIME 上的所有问题都可以用预微积分方法解决。不允许使用计算器。

factorial， noun [ C ]

MATHEMATICS   specializedUK  /fækˈtɔː.ri.əl/ US  /fækˈtɔːr.i.əl/ the product of (= the result of multiplying) a whole number and all the whole numbers below it