Question

The number of all possible selections of one or more questions from 10 given questions, each question having an alternative is
A) ${3^{10}}$
B) ${2^{10}} - 1$
C) ${3^{10}} - 1$
D) ${2^{10}}$

Answer 1

Hint:
Here, we will solve the problem by using the concept of Combinations. The combination is defined as the method of selecting an element from a set of elements such that the order of selection of elements doesn’t matter. We will form an equation by using the given condition and simplify it using the binomial expansion. We will solve it further to get the required answer.

Complete step by step solution:
We are given a total of 10 questions with an alternative.
So, the number of possible selections of one or more questions can be represented as $y$.
Thus, we get
$y = {}^{10}{C_1} \times 2 + {}^{10}{C_2} \times {2^2} + {}^{10}{C_3} \times {2^3} + {}^{10}{C_4} \times {2^4} + ........ + {}^{10}{C_{10}} \times {2^{10}}$
Now, by using the binomial expansion, we get
$ \Rightarrow {\left( {1 + x} \right)^{10}} = {}^{10}{C_0}{x^0} + {}^{10}{C_1}{x^1} + {}^{10}{C_2}{x^2} + {}^{10}{C_3}{x^3} + {}^{10}{C_4}{x^4} + ........ + {}^{10}{C_{10}}{x^{10}}$
By substituting $x = 2$ in the above equation, we get
$ \Rightarrow {\left( {1 + 2} \right)^{10}} = {}^{10}{C_0}\left( {{2^0}} \right) + {}^{10}{C_1}\left( {{2^1}} \right) + {}^{10}{C_2}\left( {{2^2}} \right) + {}^{10}{C_3}\left( {{2^3}} \right) + {}^{10}{C_4}\left( {{2^4}} \right) + ........ + {}^{10}{C_{10}}\left( {{2^{10}}} \right)$
By simplifying the equation, we get
$ \Rightarrow {\left( 3 \right)^{10}} = 1 + y$
By rearranging the terms of the above equation, we get
$ \Rightarrow y = {\left( 3 \right)^{10}} - 1$
Therefore, the possible number of selections is ${3^{10}} - 1$

Thus Option (C) is the correct answer.

Note:
We are given a total of 10 questions.
Each question selected has three possible selections such that the question may be selected or the question’s alternative may be selected or the question may be rejected.
So, the number of all possible selections such that the question may be selected or the question’s alternative may be selected or the question may be rejected is ${3^n}$ where $n$ denotes the number of questions.
Since we are given a total of 10 questions, thus the number of all possible selections such that the question may be selected or the question’s alternative may be selected or the question may be rejected is ${3^{10}}$
Number of possible selections with a question with three alternatives $ = {3^{10}}$
We are given that it is enough to select 9 questions from 10 questions.
$ \Rightarrow $ Number of possible selections with a question with three alternatives $ = {3^{10}} - 1$
Therefore, the number of all possible selections of one or more questions from 10 given questions, each question having an alternative is ${3^{10}} - 1$.