Probability
Dice and counting outcomes
Grade 12
Question:
<p>Three dice are thrown. The probability of getting a sum which is a perfect square, is</p>
<p>(a) \(\frac{2}{5}\)</p>
<p>(b) \(\frac{9}{20}\)</p>
<p>(c) \(\frac{1}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Find all possible sums that are perfect squares when throwing three dice, then count favorable outcomes.
<p><strong>Solution:</strong> $n(S) = 6 \times 6 \times 6 = 216$ (total number of ways)</p><p>The sum of numbers on three dice varies from 3 to 18. Perfect squares in this range are: 4, 9, and 16.</p><p>To find the number of favorable outcomes, we need to count the ways to get sums of 4, 9, and 16.</p><p>$n(E) = \text{Coefficient of } x^4 \text{ in } (x + x^2 + \cdots + x^6)^3 + \text{Coefficient of } x^9 + \text{Coefficient of } x^{16}$</p><p><strong>Required probability:</strong> The answer is (d) None of these.</p>
Correct Answer: D