<p>The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is</p>
Step-by-Step Solution
Key Concept: Use complementary counting: Total ways to select 4 numbers minus the ways that include at least one set of 4 consecutive numbers. Alternatively, model this as placing 4 selected numbers among 30 positions such that no two selected positions form a complete consecutive block of 4.
<p><strong>Step 1:</strong> Total ways to select 4 numbers from 1 to 30 = C(30,4) = 27405</p><p><strong>Step 2:</strong> Count selections containing at least one set of 4 consecutive numbers. If 4 consecutive numbers {n, n+1, n+2, n+3} are selected, then n can range from 1 to 27 (27 possible blocks).</p><p><strong>Step 3:</strong> For each block of 4 consecutive numbers (say {n, n+1, n+2, n+3} where n ∈ {1,2,...,27}), we've already selected 4 numbers, so there are no additional selections needed. Thus there are exactly 27 ways to select 4 consecutive numbers.</p><p><strong>Step 4:</strong> Check for overlaps: Two different blocks of 4 consecutive numbers cannot share all 4 elements (they differ by position). Blocks like {1,2,3,4} and {2,3,4,5} are different selections, so no overlap issue in complementary counting.</p><p><strong>Step 5:</strong> Number of valid selections = C(30,4) - 27 = 27405 - 27 = <strong>27378</strong></p><p>∴ Answer: A</p>
Correct Answer: A