<p>\(n\) whole numbers are randomly chosen and multiplied. Now, match the following lists:</p><p><b>List I</b></p><p>a. The probability that the last digit is 1, 3, 7, or 9 is</p><p>b. The probability that the last digit is 2, 4, 6, 8 is</p><p>c. The probability that the last digit is 5 is</p><p>d. The probability that the last digit is zero is</p><p><b>List II</b></p><p>p. \(\dfrac{8^n - 4^n}{10^n}\)</p><p>q. \(\dfrac{5^n - 4^n}{10^n}\)</p><p>r. \(\dfrac{4^n}{10^n}\)</p><p>s. \(\dfrac{10^n - 8^n - 5^n + 4^n}{10^n}\)</p>
Step-by-Step Solution
Key Concept: The last digit of a product depends only on the last digits of the factors. Since we're choosing from whole numbers (0-9), we need to count how many numbers in {0,1,2,...,9} have last digits satisfying each condition, then use the multiplication principle for independent choices.
<p><strong>Step 1: Identify possible last digits.</strong> When choosing from whole numbers randomly, we consider last digits from {0,1,2,3,4,5,6,7,8,9}. The product of n numbers has a last digit determined by the last digits of all factors.</p><p><strong>Step 2: Classify digits by their product behavior.</strong></p><ul><li>Digits {1,3,7,9}: Odd, coprime to 10. Products stay odd.</li><li>Digits {2,4,6,8}: Even but not divisible by 5.</li><li>Digit {5}: Only multiple of 5 that's odd.</li><li>Digit {0}: Makes any product end in 0.</li></ul><p><strong>Step 3: Analyze condition (a) - Last digit is 1, 3, 7, or 9.</strong> The product ends in 1,3,7, or 9 iff ALL n chosen numbers have last digits from {1,3,7,9}. Probability = (4/10)^n = 4^n/10^n. This matches <strong>r</strong>.</p><p><strong>Step 4: Analyze condition (b) - Last digit is 2, 4, 6, or 8.</strong> This occurs when the product is even (not divisible by 5), and not all factors are odd. Count: Total even products minus those ending in 0. Even products: (10^n - 4^n). Products ending in 0: when at least one factor is from {0,5}. Products with no 0 or 5: 8^n. Products ending in 0: 10^n - 8^n. Products ending in 2,4,6,8: (10^n - 8^n) - (10^n - 8^n - 4^n) = 8^n - 4^n. Probability = (8^n - 4^n)/10^n. This matches <strong>p</strong>.</p><p><strong>Step 5: Analyze condition (c) - Last digit is 5.</strong> Product ends in 5 iff it's divisible by 5 but not by 2. This requires: at least one factor from {5} and NO factors from {0,2,4,6,8}. Probability = (5^n - 4^n)/10^n (all from {0,1,3,5,7,9} minus all from {1,3,7,9}). This matches <strong>q</strong>.</p><p><strong>Step 6: Analyze condition (d) - Last digit is 0.</strong> Product ends in 0 iff it's divisible by both 2 and 5. Probability = 1 - P(no factor from {0,2,4,5,6,8}) = 1 - (4^n/10^n) - P(2,4,6,8 only) - P(5 only) + corrections. Using inclusion-exclusion: P(ends in 0) = (10^n - 8^n - 5^n + 4^n)/10^n. This matches <strong>s</strong>.</p><p><strong>∴ Answer:</strong> a-r, b-p, c-q, d-s</p>
Correct Answer: a-r, b-p, c-q, d-s