Question

In an election, the number of candidates is 1 greater than the person to be elected. If a voter can vote 254 ways, the number of candidates is
(a) 7
(b) 10
(c) 8
(d) 6

Answer 1

Hint: Here, we will assume the number of candidates as n and person to be elected as $n-1$ . Then we can say that he/she can vote for any 1 person, or any 2 person, or any 3 person up-to $n-1$ . He/she cannot vote to n candidates. So, from this series of combinations will be formed which will be equal to value ${{2}^{n}}$ and then this equation we have to equate it with 254. Thus, we will get the value of n by solving.

Complete step-by-step answer:
In the question, we are given that the number of candidates is 1 greater than the person to be elected. So, we will assume that the number of candidates to be n and the person to be elected as $n-1$ .
Now, we can say that a voter can vote in 254 ways i.e. he/she can vote any 1 person, or any 2 person, or any 3 person up-to $n-1$ . He/she cannot vote to n candidates.
So, this in mathematical form can be written as
${}^{n}{{C}_{1}}+{}^{n}{{C}_{2}}+{}^{n}{{C}_{3}}+.....{}^{n}{{C}_{n-1}}$
Now, we will add two terms here i.e. ${}^{n}{{C}_{0}}$ and ${}^{n}{{C}_{n}}$ . Also, we will subtract this in series. So, we get
${}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+{}^{n}{{C}_{2}}+{}^{n}{{C}_{3}}+.....{}^{n}{{C}_{n-1}}+{}^{n}{{C}_{n}}-{}^{n}{{C}_{0}}-{}^{n}{{C}_{n}}$
Now, we know that ${}^{n}{{C}_{0}}+{}^{n}{{C}_{1}}+{}^{n}{{C}_{2}}+{}^{n}{{C}_{3}}+.....{}^{n}{{C}_{n-1}}+{}^{n}{{C}_{n}}={{2}^{n}}$ and ${}^{n}{{C}_{0}},{}^{n}{{C}_{n}}=1$ So, putting this value in the expression, we get
${{2}^{n}}-1-1={{2}^{n}}-2$
Now, it is given that there are a total 254 ways to elect the candidate. So, we can write it as
${{2}^{n}}-2=254\Rightarrow {{2}^{n}}=256$
We know that multiplying 2 eight times we get value 256 so, we get value of n as
${{2}^{8}}=256\Rightarrow n=8$
Thus, we have assumed the number of candidates as n which is 8.
Hence, option (c) is correct.

Note: Another approach is by assuming an elected person to be n so, the number of candidates will be 1 greater than n i.e. $n+1$ . So, voters have 2 choices to elect the person. Thus, there are ${{2}^{n+1}}$ choices. Now, at least one candidate is to be selected and all candidates cannot be voted. So, we will subtract 2 from the choices, we will get as
${{2}^{n+1}}-2=254$
On solving, we get
${{2}^{n+1}}=256={{2}^{8}}$
$n+1=8\Rightarrow n=7$
Thus, we get the number of elected people as 7 but we have assumed the number of candidates as $n+1$ so, we get the answer as $n=7+1=8$ . So, the answer will be the same by this method.