<p>At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is</p>
Step-by-Step Solution
Key Concept: A voter can vote for 1, 2, 3, or 4 candidates (at least 1, not more than 4 to be elected). Sum all combinations: C(10,1) + C(10,2) + C(10,3) + C(10,4), then subtract cases where voter doesn't vote at all.
<p><strong>Step 1:</strong> Identify the constraint: voter must vote for at least 1 candidate, but not more than 4 (the number to be elected).</p><p><strong>Step 2:</strong> Count ways to vote for exactly k candidates where k ∈ {1, 2, 3, 4}:</p><p>Number of ways = C(10,1) + C(10,2) + C(10,3) + C(10,4)</p><p><strong>Step 3:</strong> Calculate each term:</p><p>C(10,1) = 10</p><p>C(10,2) = 45</p><p>C(10,3) = 120</p><p>C(10,4) = 210</p><p><strong>Step 4:</strong> Sum all cases:</p><p>Total = 10 + 45 + 120 + 210 = <strong>385</strong></p><p><strong>Alternative approach:</strong> Total ways to select any subset from 10 candidates = 2<sup>10</sup> = 1024. Subtract the case where no one is voted for (1 way) and cases where more than 4 are voted for [C(10,5) + C(10,6) + ... + C(10,10) = 2<sup>10</sup>/2 - (C(10,4) + C(10,3) + C(10,2) + C(10,1) + C(10,0))/2 by symmetry]. This gives 1024 - 1 - 638 = 385.</p><p>∴ Answer: <strong>385</strong></p>
Correct Answer: C