Question

There are 6 multiple choice questions on an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three 5 each?

Answer 1

We here have been given 6 MCQs with the first three having 4 choices each and the next 3 having 5 and we have to tell the total number of possible sequences of answers. For this, we will consider it the same as calculating the number of ways to do 6 different tasks simultaneously. The number of ways of answering each question will be equal to the number of choices of that question. Thus, the total number of sequences will come out as their product, hence, we will obtain our answer.

Complete step by step answer:
Here, we have been given 6 MCQs in which the first three have 4 choices each and the next three have 5.
Now, every question is independent of the other. Thu, the number of ways any question can be answered is equal to the number of choices the question has.
Hence, the number of ways the first three questions can be answered= 4
Similarly, the number of ways the next three questions can be answered= 5
Now, we have to tell the number of sequences in which these can be answered. This is the same as answering them simultaneously and we know that when different tasks are to be done simultaneously, the total number of ways of doing those is the product of the number of ways of performing those tasks individually.
Thus, here the total number of ways of answering the 6 MCQs is given as:
$4\times 4\times 4\times 5\times 5\times 5$
Solving this, we get:
$\begin{aligned} & 4\times 4\times 4\times 5\times 5\times 5 \\\ & \Rightarrow 64\times 125 \\\ & \therefore 8000 \\\ \end{aligned}$

Thus, the total number of sequences of answers that can be formed are 8000.

Note: Always remember when two or more tasks are to be done simultaneously (the case is of “and” or “intersection”), the number of ways of doing the different tasks are multiplied and when any one of the multiple tasks is to be done (the case of “or” or “union”), the number of doing the different tasks are added.