<p>There are 5 mangoes and 4 apples. In how many different ways can a selection of fruits be made if (i) fruits of the same kind are different (ii) fruits of the same kind are identical?</p><p>(i) Find the number of ways if fruits of the same kind are different.</p>
Step-by-Step Solution
Key Concept: When fruits of the same kind are different (distinguishable), each fruit is a distinct object. For each of the 9 fruits, we have 2 choices (select or don't select), giving 2^9 total subsets. Subtract 1 to exclude the empty selection.
<p><strong>Step 1:</strong> Understand the setup - There are 5 distinct mangoes and 4 distinct apples (total 9 distinguishable fruits).</p><p><strong>Step 2:</strong> Apply the fundamental principle - For EACH fruit, we have exactly 2 choices: either include it in the selection or exclude it.</p><p><strong>Step 3:</strong> Count total selections - By the multiplication principle: 2 × 2 × 2 × ... (9 times) = 2^9 total possible selections (including selecting no fruits).</p><p><strong>Step 4:</strong> Exclude empty selection - Since we need at least one fruit selected (a 'selection' implies non-empty): 2^9 - 1.</p><p><strong>Verification:</strong> This includes: all 1-fruit selections C(9,1), all 2-fruit selections C(9,2), ..., all 9-fruit selections C(9,9) = [C(9,1) + C(9,2) + ... + C(9,9)] = 2^9 - 1 ✓</p><p>∴ Answer: <strong>2^9 - 1 = 511</strong></p>
Correct Answer: 2^9 - 1