Selection With At Least Condition

Counting selections under an at-least condition is done by summing the valid cases, a standard JEE Advanced casework approach. With 5 red and 4 blue balls, choosing 4 balls with at least 2 red means exactly 2, 3, or 4 red balls. For exactly 2 red and 2 blue: C(5,2)·C(4,2) = 10·6 = 60. For exactly 3 red and 1 blue: C(5,3)·C(4,1) = 10·4 = 40. For exactly 4 red and 0 blue: C(5,4)·C(4,0) = 5·1 = 5. Summing gives 60 + 40 + 5 = 105. Option 100 miscounts one term. Option 120 overcounts the cases. Option 60 includes only the exactly-2-red case. Hence the number of selections is 105. Plausibility check: the total ways to pick any 4 balls from 9 is C(9,4) = 126, and subtracting the cases with fewer than 2 red, namely 0 red C(4,4) = 1 and 1 red C(5,1)C(4,3) = 5·4 = 20, gives 126 - 21 = 105, confirming the casework sum. At-least conditions are resolved either by summing the qualifying exact cases or, often more swiftly, by subtracting the few forbidden configurations from the unrestricted total. Choosing the shorter route, complementary counting when the excluded cases are fewer, is a hallmark of efficient combinatorial reasoning and a deliberate decision point these selection problems are constructed to reward.