There are 4 boxes colored red, yellow, green and blue. If 2 boxes are…

Concept

For an 'at least one of two categories' selection, use the complement principle: favourable ways = total ways minus ways where the condition fails. For selecting r items from n without repetition, use the combination count C(n, r) = n! / (r!*(n-r)!).

Application

1. Total ways to select 2 boxes out of the 4 available = 4C2 = 6.

2. Group the boxes as Green, Red, and 'Others' (Yellow, Blue - 2 boxes). The condition fails only when neither Green nor Red is chosen, i.e. both selected boxes come from 'Others': 2C2 = 1.

3. So, favourable ways = Total minus Failing = 6 minus 1 = 5.

Cross-check

Verify by splitting into the three mutually exclusive cases that satisfy the condition:

1. Green with one of the Others: 1C1 times 2C1 = 1 times 2 = 2

2. Red with one of the Others: 1C1 times 2C1 = 1 times 2 = 2

3. Both Green and Red together: 1C1 times 1C1 = 1

Adding these cases: 2 + 2 + 1 = 5, matching the complement method and confirming the result.