(a) 求单词 ALGEBRAIC 中 9 个字母的不同排列数。

(b) 求单词 ALGEBRAIC 中 9 个字母的不同排列数，要求两个 A 之间不超过两个字母。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

9个字母中只有A重复两次。

按两A之间有0、1、2个字母分类。固定左A位置后，其余七个不同字母可任意排列。

If there are $r$ letters between the As, the left A can occupy $8-r$ positions. Once both A positions are fixed, the other seven distinct letters have $7!$ arrangements.

Thus

另一条路线先选出两A中间的有序字母，再把两A及中间字母看作一个块。

Classify by the number of letters between the As.

• No letter: the AA block and seven other letters have $8!$ arrangements.
• One letter: choose that letter in 7 ways, then arrange the A_A block and six others in $7!$ ways.
• Two letters: choose and order them in $7\times6$ ways, then arrange the A__A block and five others in $6!$ ways.

Hence

也可以用(a)的全部排列减去两A间隔3至7个字母的情况。

For $r=3,4,5,6,7$ letters between the As, there are $(8-r)7!$ arrangements. Therefore the required number is