There are 3 sections in a question paper each containing 5 questions. A candidate has to solve only 5 questions, choosing at least one question from each section. In how many ways can he make his choice?
\[{}^{15}{{C}_{5}}\]
\[{}^{3}{{C}_{1}}\times {}^{12}{{C}_{4}}\] ​
2250
None of these

- Hint: In such a question, we will first find all those ways in which the candidate can perform the task he has to complete.
We have to find the number of ways in which the choices can be made.
The formula for choosing r things from n given things is as follows
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
The formula for arranging n different things is as follows
\[=n!\]
Another important formula that is a variation of the above formula is to arrange n things out of which r things are identical.
\[=\dfrac{n!}{r!}\]

Complete step-by-step solution -
In this particular question, we will first find all those ways in which the candidate can do the paper or can do 5 questions from the 3 sections in total.
As mentioned in the question, we have to find the number of ways in which a choice can be made to select 5 questions that are to be done out of the 15 questions that are equally present in 3 different sections.
Now, the choices can be made in the following ways
(2,2,1);(2,1,2);(1,2,2);(3,1,1);(1,3,1);(1,1,3)
Now, we can choose the questions in the above mentioned pattern as follows
     \[\begin{align}
  & ={}^{5}{{C}_{2}}\times {}^{5}{{C}_{2}}\times {}^{5}{{C}_{1}}+{}^{5}{{C}_{1}}\times {}^{5}{{C}_{2}}\times {}^{5}{{C}_{2}}+{}^{5}{{C}_{2}}\times {}^{5}{{C}_{2}}\times {}^{5}{{C}_{1}}+{}^{5}{{C}_{1}}\times {}^{5}{{C}_{1}}\times {}^{5}{{C}_{3}}+{}^{5}{{C}_{1}}\times {}^{5}{{C}_{3}}\times {}^{5}{{C}_{1}}+{}^{5}{{C}_{3}}\times {}^{5}{{C}_{1}}\times {}^{5}{{C}_{1}} \\
 & =3\cdot \left( {}^{5}{{C}_{2}}\times {}^{5}{{C}_{2}}\times {}^{5}{{C}_{1}}+{}^{5}{{C}_{3}}\times {}^{5}{{C}_{1}}\times {}^{5}{{C}_{1}} \right) \\
 & =3\times750 \\
 & =2250 \\
\end{align}\]
Hence, the total number of ways to do the task is 2250.

Note: - In such a question, we will first find all those ways in which the candidate can perform the task he has to complete.
The students can make an error if they don’t know about the basic formulas of permutation and combination which are given in the hint as follows
The formula for choosing r things from n given things is as follows
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\]
The formula for arranging n different things is as follows
\[=n!\]
Another important formula that is a variation of the above formula is to arrange n things out of which r things are identical.
\[=\dfrac{n!}{r!}\]