A batch of 10 cupcakes consists of 5 chocolate , 3 vanilla , and 2 strawberry . If 4 cupcakes are selected to be put into a gift box, find the number of different ways they can be chosen if the selection must include at least 2 chocolate , at most 1 vanilla , and exactly 1 strawberry cupcake.

Step-by-step solution

Let C , V , and S be the number of chocolate, vanilla, and strawberry cupcakes selected, respectively. We are given the conditions: C + V + S = 4 C 2 V 1 S = 1 Since S = 1 , we have C + V = 3 . The possible values for (C, V) satisfying C 2 and V 1 are: Case 1: C = 2 , V = 1 Case 2: C = 3 , V = 0 For Case 1 ( 2 chocolate, 1 vanilla, 1 strawberry): Number of ways = ⁵C₂ ³C₁ ²C₁ = 10 3 2 = 60 For Case 2 ( 3 chocolate, 0 vanilla, 1 strawberry): Number of ways = ⁵C₃ ³C₀ ²C₁ = 10 1 2 = 20 Total number of ways = 60 + 20 =