Question

In an election the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways, then the number of candidates is?

• A. $7$
• B. $10$
• C. $8$
• D. $6$

Answer 1

The correct option is C

$8$

Explanation for the correct option.

Finding the number of candidates:

Let the number of candidates be $n$. So,

$$\begin{gathered} {}^{n}C_1+{}^{n}C_2+{}^{n}C_3+....+{}^{n}C_{n-1}=254 \\ \Rightarrow 2^n-2=254\left[by{}^{n}C_1+{}^{n}C_2+{}^{n}C_3+....+{}^{n}C_n=2^n-1\right] \\ \Rightarrow 2^n=256 \\ \Rightarrow 2^n=2^8 \\ \Rightarrow n=8 \end{gathered}$$

Hence, option C is correct.