Question

In how many ways can 4 books of mathematics and 3 books of English can be placed on a shelf so that books on the same subject always remain together?

Answer 1

Hint: There are two possible cases that either maths books are placed on left and English books are placed on right or the English books are placed on left and maths books are placed on right. Also, we know that arrangement of n different things is equal to n! and the value of n! is given by,
n!\[=n\left( n-1 \right)\left( n-2 \right).\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. 3}\times \text{2}\times \text{1}\]

Complete step-by-step answer:
We have been asked to find the number of ways that 4 books of mathematics and 3 books of English can be placed on a shelf so that books on the same subject always remain together.
Let the 4 mathematics books are \[{{M}_{1}},{{M}_{2}},{{M}_{3}},{{M}_{4}}\] and 3 English books are \[{{E}_{1}},{{E}_{2}}\text{ and }{{\text{E}}_{\text{3}}}\]
Now there are two possible cases: maths books are placed on left and English books are placed on right or vice versa.
Case I: \[\left. {{M}_{1}}\,{{M}_{2}}\,{{M}_{3}}\,{{M}_{4}} \right|{{E}_{1}}\,{{E}_{2}}\,{{\text{E}}_{\text{3}}}\]
Case II: \[\left. {{E}_{1}}\,{{E}_{2}}\,{{\text{E}}_{\text{3}}} \right|{{M}_{1}}\,{{M}_{2}}\,{{M}_{3}}\,{{M}_{4}}\]
We know that arrangement of n things \[=\text{n}!\]
For Case I:
Arrangement of Math’s books = 4!
\[\begin{align}
  & \Rightarrow 4\times 3\times 2\times 1 \\
 & \Rightarrow 24 \\
\end{align}\]
Arrangement of English books = 3!
\[\begin{align}
  & \Rightarrow 3\times 2\times 1 \\
 & \Rightarrow 6 \\
\end{align}\]
So, possible ways in this case \[\Rightarrow 24\times 6\]
\[\Rightarrow 144\]
For Case II:
Arrangement of Math’s books \[\Rightarrow \text{4}!=\text{24}\]
Arrangement of English books \[\Rightarrow \text{3}!=\text{6}\]
So, possible way in this case \[\Rightarrow 24\times 6\]
\[\Rightarrow 144\]
Hence, total possible ways that books of the same subject always remain together \[\Rightarrow 144+144\text{ = 288}\]

Note: One common mistake that we generally do is to consider only one case and forget about the second case then we get the incorrect answer. So, it is better to assume some variables and visualize the problem according to the question to avoid such type of mistakes.