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

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(C) Let the number of candidates be $n$.
Since the number of candidates is $1$ more than the number of persons to be elected,the number of persons to

Vedclass · Accepted Answer

$8$

Vedclass · Answer

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

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)

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