Question

In an election, the number of candidates is one more than the number of seats. If a voter can cast his vote in 30 ways, find the number of candidates (when a voter can cast his vote for one or more seats).

• A. \(31\)
• B. \(29\)
• C. \(5\)
• D. \(6\)

Hint:

A frequently used identity: \[ \sum_{r=1}^{n-1} {n \choose r} = 2^n-2 \] obtained by removing the empty set and the whole set from all subsets.

Answer 1

The Correct Option is D

Concept: If there are \(n\) candidates and \(n-1\) seats, a voter may vote for any non-empty subset of candidates having at most \(n-1\) members. Thus total voting ways are \[\begin{aligned} \sum_{r=1}^{n-1} {n \choose r} \end{aligned}\] Using \[\begin{aligned} \sum_{r=0}^{n} {n \choose r} = 2^n \end{aligned}\] we get \[\begin{aligned} \sum_{r=1}^{n-1} {n \choose r} = 2^n-2 \end{aligned}\]

Step 1: Use the given condition. \[\begin{aligned} 2^n-2=30 \end{aligned}\] \[\begin{aligned} 2^n=32 \end{aligned}\] \[\begin{aligned} n=5 \end{aligned}\] This gives number of candidates \[\begin{aligned} n+1=6 \end{aligned}\]

Step 2: Determine the number of candidates. \[\begin{aligned} \boxed{6} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.