Question

There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible (correct or incorrect) answers are there to this question?

Answer 1

Hint: For solving this question, first we will see the concept of the fundamental principle of multiplication. After that, we will find the number of choices available for each element in column A, when we assign them elements from the six elements of column B in succession. Then, we will multiply all the number of choices to get the final answer.

Complete step-by-step answer:
It is given that in a question, there are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. And we have to find the total number of ways possible (correct or incorrect) answers are there to this question.
Now, before we proceed we should know the following important concept and formulas:
Fundamental Principle of Multiplication:
If there are two jobs such that one of them can be completed in $m$ ways, and when it has been completed in any of these $m$ ways, the second job can be completed in $n$ ways. Then, two jobs in succession can be completed in $m\times n$ ways.
Now, let ${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}},{{a}_{5}},{{a}_{6}}$ are the six elements of column A.
Now, we will find the number of choices available for each element in column A, when we assign them elements from the six elements of column B in succession.
Now, as for the first element of column A, there will be six elements of column B and subsequently, the number of choices will decrease by $1$ as we move further. Then,
The number of choices for the first element ${{a}_{1}}=6$ choices.
The number of choices for the second element ${{a}_{2}}=6-1=5$ choices.
The number of choices for the third element ${{a}_{3}}=6-2=4$ choices.
The number of choices for the fourth element ${{a}_{4}}=6-3=3$ choices.
The number of choices for the fifth element ${{a}_{5}}=6-4=2$ choices.
The number of choices for the sixth element ${{a}_{6}}=6-5=1$ choices.
Now, from the fundamental principle of multiplication, we can say that the total required number of ways will be equal to $6\times 5\times 4\times 3\times 2\times 1=720$ ways.
Thus, the total number of ways possible (correct or incorrect) answers are there to this question will be equal to $720$ ways.

Note:Here, the student should first understand what is asked in the question and then proceed in the right direction to get the correct answer quickly. After that, we should apply the fundamental principle of counting very carefully. Moreover, as the number of elements in column A and B are equal, we could have directly used the formula $n!$ to get the total number of ways with the value of $n=6$ .