Question

A student has to sell $2$ books from a collection of $6$ math, $7$ science, and $4$ economics books. How many choices are possible if

(a) both books are to be on the same subject?

(b) the books are to be on different subjects?

Short answer

(a) The possible number of choices if both books are to be on the same subject is $42$.

(b) The possible number of choices if both books are to be on the different subjects is$94$.

Step-by-step solution

Part (a) Step 1. Given information.

Number of math books $=6$

Number of science books role="math" localid="1647785628795" $=7$

Number of economics books $=4$

Number of books to be sold$=2$

Part (a) Step 2. Calculate the possible number of choices if both books are to be on the same subject.

No. of choices that both books sold are of maths $=\left(\begin{array}{c}6\\ 2\end{array}\right)=\frac{6!}{2!4!}=\frac{6\times 5\times 4!}{2\times 1\times 4!}=15$

No. of choices that both books sold are of science $=\left(\begin{array}{c}7\\ 2\end{array}\right)=\frac{7!}{2!5!}=\frac{7\times 6\times 5!}{2\times 1\times 5!}=21$

No. of choices that both books sold are of economics $=\left(\begin{array}{c}4\\ 2\end{array}\right)=\frac{4!}{2!2!}=\frac{4\times 3\times 2!}{2\times 1\times 2!}=6$

As only one of the above three choices is possible so, the possible number of choices$=15+21+6=42$

Part (b) Step 3. Given information.

Number of math books$=6$

Number of science books$=7$

Number of economics books$=4$

Number of books to be sold$=2$

Part (b) Step 4. Calculate the possible number of choices if both books are to be on the different subjects.

No. of choices if one book is of math and other book is of science $=\left(\begin{array}{c}6\\ 1\end{array}\right)\left(\begin{array}{c}7\\ 1\end{array}\right)=\frac{6!}{1!5!}\times \frac{7!}{1!6!}=\frac{6!\times 7\times 6\times 5!}{5!\times 6!}=42$

No. of choices if one book is of math and other book is of economicsrole="math" localid="1647787277434" $=\left(\begin{array}{c}6\\ 1\end{array}\right)\left(\begin{array}{c}4\\ 1\end{array}\right)=\frac{6!}{1!5!}\times \frac{4!}{1!3!}=\frac{6\times 5!\times 4\times 3!}{5!\times 3!}=24$

No. of choices if one book is of science and other book is of economics role="math" localid="1647787332330" $=\left(\begin{array}{c}7\\ 1\end{array}\right)\left(\begin{array}{c}4\\ 1\end{array}\right)=\frac{7!}{1!6!}\times \frac{4!}{1!3!}=\frac{7\times 6!\times 4\times 3!}{6!\times 3!}=28$

Therefore, the possible number of choices if both books are to be on the different subjects is$=42+24+28=94$