Problem 4. (15%) There are two apples and four oranges. You would like to buy three of them, but at least one apple should be selected. Please show that how many choices you have. Please justify your answer. Otherwise, you get 0points.

4. Let G be a simple graph. A path of G is a sequence of distinct vertices v0,v1, Uh such that and are adjacent for each i = 1,2,..., k. The length of the path v0, v1 , ..., is k. The degree of a vertex is the number of edges incident to that vertex. Show that if the minimum degree of G is greater than or equal to k, then G has a path whose length is at least k.

Problem 4. (15%) There are two apples and four oranges. You would like to buy three of them, but at least one apple should be selected. Please show that how many choices you have. Please justify your answer. Otherwise, you get 0points.

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