Permutations & Combinations
Distribution with repetition
Grade None
Question:
<p>In a shop there are five types of ice-creams available. A child buys six ice-creams.</p><p><strong>Statement-1:</strong> The number of different ways the child can buy the six ice-creams is \({}^{10}C_5\).</p><p><strong>Statement-2:</strong> The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row.</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: This is a problem of distributing identical items (6 ice-creams) into distinct categories (5 types), which is a 'stars and bars' problem solved by the formula C(n+k-1, k) where n=6 items and k=5 types. Recognize that Statement-2's arrangement of 6 A's and 4 B's represents exactly this same combinatorial structure.
<p><strong>Step 1: Identify the problem type</strong><br/>We need to find the number of ways to choose 6 ice-creams from 5 types where repetition is allowed and order doesn't matter (composition problem).</p><p><strong>Step 2: Apply stars and bars formula</strong><br/>The number of ways to distribute n identical items into k distinct groups is C(n+k-1, k-1) = C(n+k-1, n).<br/>Here: C(6+5-1, 5-1) = C(10, 4) = <sup>10</sup>C<sub>5</sub></p><p><strong>Step 3: Verify Statement-2</strong><br/>Arranging 6 A's and 4 B's in a row requires choosing 6 positions for A's out of 10 total positions = C(10,6) = C(10,4) = <sup>10</sup>C<sub>5</sub>.<br/>This is equivalent! The A's represent 'ice-creams bought' and B's represent 'dividers between types.'</p><p><strong>Step 4: Evaluate statements</strong><br/>Statement-1: TRUE (answer is <sup>10</sup>C<sub>5</sub>)<br/>Statement-2: TRUE (same answer, same underlying combinatorial structure)</p><p>∴ Answer: B (Both statements are correct and Statement-2 explains Statement-1)</p>
Correct Answer: B