Question

There are 5 multiple choice questions in the test. If the first three questions have 4 choices each and the next two have 5 choices each, the number of answers possible is
(a) 1500,
(b) 1600,
(c) 1700,
(d) 1800.

Answer 1

We first find the number of possibilities of each equation. We then look at the definitions of sum rule and product rule. We then try to categorize our situation to any one of those two rules. We now apply that rule which will be most probable for the situation to get the total number of ways, which will be the required result of the problem.

Complete step by step answer:
According to the problem, we have 5 multiple choice questions which have options as given below. We have the first 3 questions with 4 choices and next 2 questions with 5 choices.
Let us find the total no. of possibilities to answer each question.
So, we get the number of possibilities for the first question given as 4 ways.
We get the number of possibilities for the second question given as 4 ways.
We get the number of possibilities for the third question given as 4 ways.
We get the number of possibilities for the fourth question given as 5 ways.
We get the number of possibilities for the fifth question given as 5 ways.
Let us recall the Rule of sum and Rule of product.
Rule of sum: In combinatorics, the rule of sum of 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. Given P, Q works cannot be done together. Total number of ways to do both P, Q are given by $\left( A+B \right)$ 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 $A\times B$ ways.
From the above definitions as these are independent events, we use product rule.
By applying product rule here, we get the total ways as Total ways=$4\times 4\times 4\times 5\times 5={{4}^{3}}\times {{5}^{2}}$.
By simplifying the above equations, we can write the ways as Total ways =$64\times \text{25=1600}$ways.

So, the correct answer is “Option B”.

Note: We should not confuse with the total number of ways of answering the problem. We should always remember that answering a question is the number of ways choosing one of the options from the given options. We should not compare and apply sum rule in this case by which we get wrong answers while doing product.