In an election,the number of candidates is 1 | vedclass

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Question

(C) Let the number of candidates be $n$. The number of persons to be elected is $n-1$.
$A$ voter can vote for any number of candidates from $1$ to $n

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(C) Let the number of candidates be $n$. The number of persons to be elected is $n-1$.
$A$ voter can vote for any number of candidates from $1$ to $n-1$.
The total number of ways a voter can vote is given by the sum of combinations:
$^nC_1 + ^nC_2 + \dots + ^nC_{n-1} = 254$
We know that the sum of binomial coefficients is $\sum_{k=0}^{n} {^nC_k} = 2^n$.
Therefore,$^nC_0 + ^nC_1 + ^nC_2 + \dots + ^nC_{n-1} + ^nC_n = 2^n$.
Substituting $^nC_0 = 1$ and $^nC_n = 1$,we get:
$1 + (^nC_1 + ^nC_2 + \dots + ^nC_{n-1}) + 1 = 2^n$
$1 + 254 + 1 = 2^n$
$256 = 2^n$
$2^8 = 2^n$
Thus,$n = 8$.

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:

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