How many edges are there in an n-cube?
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n*2^n
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n^3
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n*2^(n-1)
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n*(n-1)
An n-dimensional cube (n-cube) has each vertex connected to n edges. There are 2^n vertices total. Each edge is counted twice (once from each endpoint), so total edges = n * 2^n / 2 = n * 2^(n-1). This gives option C.
To answer this question, you need to understand the concept of an n-cube.
An n-cube, also known as a hypercube, is a geometric shape that exists in n-dimensional space. It is formed by connecting the vertices of a (n-1)-cube to the corresponding vertices of another (n-1)-cube.
To determine the number of edges in an n-cube, we need to consider how many edges are there in each (n-1)-cube and how many edges are added when connecting the corresponding vertices.
Let's go through each option to understand why it is correct or incorrect:
Option A) n*2^n - This option is incorrect. Multiplying n by 2^n does not account for the additional edges added when connecting the corresponding vertices.
Option B) n^3 - This option is incorrect. Raising n to the power of 3 does not accurately represent the number of edges in an n-cube.
Option C) n*2^(n-1) - This option is correct. The number of edges in an n-cube is given by n multiplied by 2^(n-1). This accounts for the number of edges in each (n-1)-cube and the additional edges added when connecting the corresponding vertices.
Option D) n*(n-1) - This option is incorrect. Multiplying n by (n-1) does not account for the additional edges added when connecting the corresponding vertices.
The correct answer is Option C) n*2^(n-1). This option is correct because it accurately represents the number of edges in an n-cube.
Therefore, the correct answer is C) n*2^(n-1).