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  4. 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

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Q. 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

Permutations and Combinations

Solution:

Let there be $n$ candidates. Then
${ }^{n} C_{1}+{ }^{n} C_{2}+\cdots+{ }^{n} C_{n-1}=254 $
$\Rightarrow 2^{n}-2=254 $
or $ 2^{n}=2^{8} $ or $ n=8$