Question

In a club election, the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote is $126$, then the number of contestants is
1)$4$
2)$5$
3)$6$
4)$7$

Answer 1

Hint: To solve this question we have to find the number of contestants. Now, we have to find the number of ways one voter can vote for one candidate irrespective of any condition. Next we have to subtract the conditions not applicable according to the question. Next we will form an equation as the total number of ways a voter can vote for a candidate is given. By solving the equation, we can find the number of contestants, which is our required answer.

Complete step by step answer:
As per the given question, voters have to vote for at least $1$ candidate and for maximum $n - 1$ candidates, where $n$ is the number of candidates.
For each candidate, voters have two options: to vote or not to vote.
Hence, there would be total ${2^n}$ ways
Now, if we exclude the condition where voter will vote none or will vote to all the candidates, then,
${2^n} - 2 = 126$
Adding $2$ on both sides of the equation, we get,
$ \Rightarrow {2^n} = 126 + 2$
$ \Rightarrow {2^n} = 128$
Now, we can write it as,
$ \Rightarrow {2^n} = {2^7}$
$ \Rightarrow n = 7$
Therefore, the total number of contestants is $7$.
Thus, the correct answer is 4.

Note: In case of problems with two choices to be made about any arbitrary no. of elements, we use the no. of ways it can be done is given by ${2^n}$. As in this case it is done, the no. of ways a voter can vote for $n$ contestants is given by ${2^n}$.