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（美國數(shù)學邀請考試）是介于 AMC 10 或 AMC 12 和 USAMO 之間的中級考試。所有參加AMC 12并在滿分150分中取得100分或以上或排名前5%的學生均可應邀參加AIME考試。所有參加 AMC 10 考試并在 150 分的滿分中取得 120 分或以上，或排名前 2.5%的學生也有資格參加 AIME 考試。美國數(shù)學邀請考試不收取額外的報名費，除非您選擇參加第二次考試。前 10 名學生和超過 10 名學生需支付額外的管理費/運費。AMC 10/12 和 AIME 教師手冊對此有更詳細的解釋。

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.

考試內容

AIME 考試共 15 道題，考試時間為 3 小時，每道題的答案都是 0 到 999 之間的整數(shù)。AIME 的試題難度更大，學生很難通過猜測獲得正確答案。與 AMC 10 和 AMC 12（以及 USAMO）一樣，AIME 上的所有問題都可以用預微積分方法解決。不允許使用計算器。

factorial，?noun?[?C?]

MATHEMATICS???specializedUK??/f?k?t??.ri.?l/?US??/f?k?t??r.i.?l/?the?product?of?(= the?result?of?multiplying)?a?whole?number?and all the?whole?numbers?below it