<p>In an election, the number of candidates is one greater than the persons to be elected. If a voter can vote in 254 ways, the number of candidates is</p>
Step-by-Step Solution
Key Concept: A voter can vote by choosing any non-empty subset of candidates up to the number of positions available. If there are n candidates and r positions, the voter has C(n,1) + C(n,2) + ... + C(n,r) ways to vote, which equals 2^n - 2^(n-r) - 1 when n = r+1.
<p><strong>Step 1:</strong> Let the number of candidates = n and persons to be elected = r. Given: n = r + 1</p><p><strong>Step 2:</strong> A voter can vote for 1 candidate, or 2 candidates, or ... or r candidates. Total ways = C(n,1) + C(n,2) + ... + C(n,r) = 254</p><p><strong>Step 3:</strong> We know that C(n,0) + C(n,1) + C(n,2) + ... + C(n,n) = 2^n</p><p>By symmetry, since n = r+1: C(n,1) + C(n,2) + ... + C(n,r) = C(n,0) + C(n,1) + ... + C(n,r) = [2^n - C(n,n)] / 2 = (2^n - 1) / 2 when n = r+1</p><p><strong>Step 4:</strong> Actually, C(n,1) + C(n,2) + ... + C(n,n-1) = 2^n - 2. So for n = r+1: C(n,1) + C(n,2) + ... + C(n,r) = 2^n - 1 - 1 = 2^(n-1) + 2^(n-1) - 2 = 2^n - 2 when excluding C(n,0) and C(n,n)</p><p><strong>Step 5:</strong> Setting 2^n - 2 = 254, we get 2^n = 256 = 2^8, so n = 8</p><p>∴ Answer: C (The number of candidates is <strong>8</strong>)</p>
Correct Answer: C