Question

高中排列与组合的问题 高手解答

因为是国际学校，所以题目是英文版本的，第7题！
中文如下（可能翻译的有点怪），一个袋子里装有20个巧克力，15个太妃糖，12个薄荷糖，如果3个糖果被随机选择，那么可能性有多少？
a.全部不同 b.全是巧克力 c.全部一样 d.全部不是巧克力

谢谢了

Answer 1

a. 240/1081, approximately 22.20%.
b. 76/1081, approximately 7.03%.
c. 121/1081, approximately 11.19%.
d. 195/1081, approximately 18.04%.

First of all, the number of ways of choosing 3 sweets from these 20+15+12=47 is
{47 choose 3}=16215. We will use this number as denominators in every case.

a. All different. There are
20*15*12=3600
ways to choose one chocolate, one toffee, and one peppermint.
Thus the probability is 3600/16215=240/1081, approximately 1.48%.

b. All chocolates. There are
{20 choose 3}=1140
ways to choose three chocolates.
So the probability is 1140/16215=76/1081, approximately 7.03%.

c. All the same. Then either all chosen sweets are chocolates, or all chosen sweets are toffees, or all chosen sweets are peppermints. The number of ways to do these are respectively
{20 choose 3}=1140,
{15 choose 3}=455,
and
{12 choose 3}=220.
Therefore, the desired propability is (1140+455+220)/16215=121/1081, approximately 11.19%.

d. All not chocolates. Then we choose three sweets from non-chocolates, which has a total number 15+12=27. There are
{27 choose 3}=2925
ways to do this. So the probability is 2925/16215=195/1081, approximately 18.04%.

Answer 2

a.全部不同
[C(1,20)+C(1,15)+C(1,12)] ÷ [C(3,20+15+12)]=?

b.全是巧克力
[C(3,20] ÷ [C(3,20+15+12)]=?

c.全部一样
{[C(3,20] ÷ [C(3,20+15+12)]}+{[C(3,15] ÷ [C(3,20+15+12)]}+{[C(3,12] ÷ [C(3,20+15+12)]}=?

d.全部不是巧克力
{[C(3,20+15+12)] - [C(3,20)+C(1,20)+C(1,20)]} ÷ [C(3,20+15+12)]=?

答案自己算一下，应该没有问题了