A biscuit tin contains 4 chocolate, 2 oatmeal, and 14 ginger biscuits. If 5 biscuits are selected at random, how many selections contain exactly 2 oatmeal biscuits?

Step 1: Identify the total number of biscuits of each type and the selection criteria. There are 4 chocolate, 2 oatmeal, and 14 ginger biscuits. Total biscuits = 4 + 2 + 14 = 20. We need to select 5 biscuits, and 2 of them must be oatmeal. Step 2: Calculate the number of ways to select the oatmeal biscuits. Since there are 2 oatmeal biscuits available and we must select 2, the number of ways to do this is given by the combination formula C(n, k) = (n!)/(k!(n-k)!): C(2, 2) = (2!)/(2!(2-2)!) = (2!)/(2!0!) = (2 × 1)/((2 × 1)(1)) = 1 There is only 1 way to select 2 oatmeal biscuits from the 2 available. Step 3: Calculate the number of ways to select the remaining biscuits. We need to select a total of 5 biscuits. Since 2 oatmeal biscuits have already been selected, we need to select 5 - 2 = 3 more biscuits. These remaining 3 biscuits must come from the chocolate and ginger biscuits, as the oatmeal biscuits have been fully accounted for. The number of chocolate and ginger biscuits available is 4 + 14 = 18. The number of ways to select 3 biscuits from these 18 is: C(18, 3) = (18!)/(3!(18-3)!) = (18!)/(3!15!) = (18 × 17 × 16 × 15!)/(3 × 2 × 1 × 15!) C(18, 3) = (18 × 17 × 16)/(3 × 2 × 1) = (4896)/(6) = 816 Step 4: Calculate the total number of ways. To find the total number of ways to select the 5 biscuits with the given condition, multiply the number of ways from Step 2 and Step 3: Total ways = C(2, 2) × C(18, 3) = 1 × 816 = 816 The number of ways this can be done is 816. 3 done, 2 left today. You're making progress.