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
$\begin{align}
  & \left( a \right)1500 \\
 & \left( b \right)1600 \\
 & \left( c \right)1700 \\
 & \left( d \right)1800 \\
\end{align}$

Answer 1

Hint: First find the number of possibilities of each equation. Look at definitions of sum rule and product rule. Try to categorize our situation to any one of them. Now apply that rule to get the total number of ways. This total number of ways of answering is the required result in this equation.

Complete step-by-step answer:
Given condition in the question is written as follow:
3 questions have 4 choices and 2 questions have 5 choices.
Total question present in the test is given as 5 questions.
Number of possibilities for the first question is given as 4 ways.
Number of possibilities for the second question is given as 4 ways.
Number of possibilities for the third question is given as 4 ways.
Number of possibilities for the fourth question is given as 5 ways.
Number of possibilities for the fifth question is given as 5 ways.
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 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.
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 =$64\text{*25=1600}$ways.
Therefore option$\left( b \right)$ is correct for this question.

Note: Don’t confuse with ways. Always remember that choices are a number of ways. Generally students compare and apply sum rule in this case by which they get wrong answers while doing product look carefully you must have three 4’s, two 5’s to get the correct result.