Question

Find the total number of ways of answering five objective type questions, each question having four choices.

Answer 1

Hint: Find the categories which are present in the given equation. Find the possibilities present in each case. Now check whether you have to add the categories or multiply them to get the result of total possibilities. Here each question has independent choices. So, you must multiply all the question’s possibilities to get the final result. So, first find possibilities for each question.

Complete step-by-step answer:
Rule of Sum: - In combinatorics, the rule of sum or addition principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. P, Q words cannot be done together. Total number of ways to do both P, Q are given by (A + B) ways.
Rule of product: - In combinatorics, the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by (A.B) ways.
By listing all possible categories, we get:
Question 1, Question 2, Question 3, Question 4, Question 5.
For given 5 Questions each have 4 multiple choices.
By listing each category with their possibilities, we get:
Possibilities of answering Question 1 = 4
Possibilities of answering Question 2 = 4
Possibilities of answering Question 3 = 4
Possibilities of answering Question 4 = 4
Possibilities of answering Question 5 = 4
Here options of 1 & 2 can be answered together. Similarly to any pair of questions.
So, this comes under product rule. So, we must multiply all.
By applying product rule, we get:
Total possibilities = \[4\times 4\times 4\times 4\times 4={{4}^{5}}\]
Therefore, there are 1024 ways to answer 5 questions with 4 choices each.

Note: Be careful while categorizing into product rule or sum value. You must carefully check whether there is any dependence between 2 works or not. Students generally apply sum rule but it is wrong you must apply product rule. Generally students confuse between the number of questions and number of choices. So, read the question carefully.