<p>There are 5 mangoes and 4 apples. In how many different ways can a selection of fruits be made if fruits of the same kind are identical?</p>
Step-by-Step Solution
Key Concept: When items of the same kind are identical, we only need to count how many of each type to select (not which specific ones). For each fruit type, the number of ways is one more than the quantity available (including selecting zero).
<p><strong>Step 1:</strong> Since fruits of the same kind are identical, we only need to decide how many mangoes and how many apples to select.</p><p><strong>Step 2:</strong> For 5 identical mangoes: we can select 0, 1, 2, 3, 4, or 5 mangoes → <strong>6 ways</strong></p><p><strong>Step 3:</strong> For 4 identical apples: we can select 0, 1, 2, 3, or 4 apples → <strong>5 ways</strong></p><p><strong>Step 4:</strong> Total selections = 6 × 5 = 30 ways (including the case of selecting nothing)</p><p><strong>Step 5:</strong> Since we need at least one fruit in a valid selection, subtract the case where we select 0 mangoes and 0 apples.</p><p><strong>Step 6:</strong> Number of ways = 30 - 1 = <strong>29</strong></p>
Correct Answer: 29