A coach needs to select a 4 - player starting lineup from a pool of 10 players : 5 guards 3 forwards 2 centres Find the number of different selections if the 4-player starting lineup must include: At least 1 guard At most 1 forward Exactly 1 centre

Step-by-step solution

Let g , f , and c be the number of guards, forwards, and centres selected respectively. We are given the following conditions: g + f + c = 4 g 1 f 1 c = 1 Substituting c = 1 into the first equation, we get g + f = 3 . Since f 1 , the possible values for f are 0 and 1 . Case 1: f = 0 If f = 0 , then g = 3 . This satisfies g 1 . The number of ways to select 3 guards, 0 forwards, and 1 centre is: ⁵C₃ ³C₀ ²C₁ = 10 1 2 = 20 Case 2: f = 1 If f = 1 , then g = 2 . This satisfies g 1 . The number of ways to select 2 guards,