题目

Problem

A group of 12 people consists of 3 boys, 4 girls and 5 adults.

(a) In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?

(2)

(b) In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?

(4)

The same group of 12 people stand in a line.

(c) How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?

(4)

题目中文翻译

一个 12 人的群体由 3 名男孩、4 名女孩和 5 名成人组成。

(a) 从这群人中选出一支 5 人队伍，要求恰好有一名成人，有多少种选法？

(2)

(b) 从这群人中选出一支 5 人队伍，要求至少有 2 名男孩和至少 1 名女孩，有多少种选法？

(4)

同样的这 12 人排成一列。

(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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一成人从5中选，其余四人从7名儿童中选。

答题过程

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N=\binom51\binom74=5(35)=\boxed{175}.

(b)

解法一

思路

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男孩至少两人只可能2或3人；再列至少一女孩的五种构成。

答题过程

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Classify the team by its numbers of boys, girls and adults:

Boys	Girls	Adults	Number
$2$	$1$	$2$	$\binom32\binom41\binom52=120$
$2$	$2$	$1$	$\binom32\binom42\binom51=90$
$2$	$3$	$0$	$\binom32\binom43=12$
$3$	$1$	$1$	$\binom33\binom41\binom51=20$
$3$	$2$	$0$	$\binom33\binom42=6$

Therefore

N=120+90+12+20+6=\boxed{248}.

(c)

解法一

思路

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先按顺序选两端成人，内部十人把三男孩合为一块后为八个对象。

答题过程

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Choose and order the two adults at the ends in $5\times4=20$ ways.

Inside the line are three adults, four girls and the block of three boys: eight distinct objects, with $8!$ arrangements. The boys can be ordered inside their block in $3!$ ways. Hence

N=(5)(4)\,8!\,3!=\boxed{4838400}.

CIE 9709 2022 March Paper 52 Q5

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