<p>If any four numbers are selected and they are multiplied, then the probability that the last digit will be 1, 3, 5 or 7 is</p>
Step-by-Step Solution
Key Concept: The last digit of a product depends only on the last digits of the factors. For the product to end in 1, 3, 5, or 7 (odd digits), ALL four numbers must be odd, since any even number makes the product even.
<p><strong>Step 1:</strong> Identify digits with odd last digits: {1, 3, 5, 7} — there are 4 odd digits out of 10 total digits (0-9).</p><p><strong>Step 2:</strong> For a product to have last digit 1, 3, 5, or 7, the product must be odd. A product is odd if and only if ALL factors are odd.</p><p><strong>Step 3:</strong> Probability that one selected number is odd = 4/10 = 2/5.</p><p><strong>Step 4:</strong> Since we select 4 independent numbers, probability that all four are odd = (2/5)⁴ = 16/625.</p><p><strong>Step 5:</strong> Verify: If all four numbers are odd, their product is odd, ending in one of {1, 3, 5, 7}. If any number is even, the product is even.</p><p>∴ Answer: <strong>16/625</strong> or <strong>(2/5)⁴</strong></p>
Correct Answer: C