跪求一排列组合题解释～

yongyang86r

32 How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?
a)21
b)170
c)340
d)357
e)420

n(n-3)/2 is the total number of diagonals

Each vertex sends out n-3 diagonals (n being the number of vertices)

so, to answer this question above,

n(n-3)/2 - (x*n-3 ) = (n-3) [n/2 - x]

substitute n with number of sides (which is equal to number of vertices and x as the number of vertices that do not connect)

32 => 18 * [21/2 - 1] = 9*19 = 171 (??? answer choice does not exist??)

Help～～

lanseb8ma

use the formula you have provided, if every vertice is connected to the rest of the group, we could get:21*(21-3)/2=189 diagonals. But the question says one of the vertices is not connected to any diagonal. Whichever vertice it is,the situation reduces (21-3)=18 diagonals: 21 sides polygon has 21 vertices, 1 vertice should have 18 diagonals leading to all other vertices except its two adjacent vertices and itself (which add to 3).

Therefore, the answer should be (189-18)=171,would it be possible that the given option is wrong?

yongyang86r

以下是引用lanseb8ma在2007-8-2 14:35:00的发言：

use the formula you have provided, if every vertice is connected to the rest of the group, we could get:21*(21-3)/2=189 diagonals. But the question says one of the vertices is not connected to any diagonal. Whichever vertice it is,the situation reduces (21-3)=18 diagonals: 21 sides polygon has 21 vertices, 1 vertice should have 18 diagonals leading to all other vertices except its two adjacent vertices and itself (which add to 3).

Therefore, the answer should be (189-18)=171,would it be possible that the given option is wrong?

That is what i thought! Maybe it is wrong.But the official answer is:

The best answer is B.

We have 20 vertices linking to 17 others each: that is 17*20=340. We divide that by 2 since every diagonal connects two vertices. 340/2=170. The vertex that does not connect to any diagonal is just not counted.

annika_shao

能不能用中文解释一下，题目是什么意思？有些不太明白

Factiva

The right answer should be 171. The "official" explanation illogically turns the question into a polygen with 20 vertices, missing one diagonal.

ll_422

We have 20 vertices linking to 17 others each: that is 17*20=340. We divide that by 2 since every diagonal connects two vertices. 340/2=170. The vertex that does not connect to any diagonal is just not counted.

---- that's  right.

---- that's  right.

yongyang86r

以下是引用Factiva在2007-8-2 22:15:00的发言：
The right answer should be 171. The "official" explanation illogically turns the question into a polygen with 20 vertices, missing one diagonal.

The "20" in the official answer means the left 20 vertices, not misunderstanding the 21 vertices to 20 vertices.

sheilajin

以下是引用Factiva在2007-8-2 22:15:00的发言：
The right answer should be 171. The "official" explanation illogically turns the question into a polygen with 20 vertices, missing one diagonal.

AGREE!

170没有算那个不连接任何diagonal那个顶点的相邻两个点连成的那条diagonal

yongyang86r

So the official answer must be wrong...