题目

Problem

A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.

(a) In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?

(4)

The 10 members of the club stand in a line for a photograph.

(b) How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?

(2)

For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club.

(c) How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?

(4)

题目中文翻译

一个新成立的乡村社交俱乐部有 10 名会员，其中 6 名男性、4 名女性。俱乐部委员会将由 5 名会员组成。

(a) 如果委员会必须包括至少 2 名男性和至少 1 名女性，选出这个 5 人委员会共有多少种方法？

(4)

俱乐部的 10 名会员站成一排拍照。

(b) 如果所有男性站在一起，所有女性也站在一起，这 10 名会员共有多少种不同的排列？

(2)

拍摄第二张照片时，会员站成两排，后排 6 人，前排 4 人。Olly 和他的姐妹 Petra 是俱乐部的两名会员。

(4)

原卷通用作答要求

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。

解答

(a)

解法一

思路

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五人委员会且至少两男一女，人数构成只有4男1女、3男2女、2男3女。

答题过程

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\begin{align*} N=&\,\binom64\binom41+\binom63\binom42\\[4mm] &\,\hspace{2pt}+\binom62\binom43\\[4mm] =&\,60+120+60=\boxed{240}. \end{align*}

(b)

解法一

思路

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男女各成一块，两块有2种次序，块内再分别排列。

答题过程

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N=2!\times6!\times4!=\boxed{34560}.

(c)

解法一

思路

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前排相邻位置对有3个，兄妹顺序2种，其余8人可填剩余8个指定位置。

答题过程

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There are three adjacent pairs of positions in the front row. Choose one for Olly and Petra, and arrange them in 2 ways. Arrange the remaining eight people in the eight remaining positions in $8!$ ways. Thus

N=3\times2\times8!=\boxed{241920}.

解法二

思路

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也可以先选另外两位前排成员，把兄妹作为块安排前排，再排后排六人。

答题过程

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Choose the other two front-row members in $\binom82$ ways. Treat Olly and Petra as a block: this block and the two others have $3!$ arrangements, with 2 internal orders for the block. The six remaining people have $6!$ back-row arrangements. Therefore

N=\binom82\times3!\times2\times6!=\boxed{241920}.

CIE 9709 2024 March Paper 52 Q6

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程

解法二

思路

答题过程