Question

In a shelf there are 10 English and 8 Telugu books. The number of ways in which 6 books can be chosen if a particular English book is excluded and a particular Telugu book is excluded is:
(a) $${}^9{C}_3\cdot {}^7{C}_3$$
(b) $${}^{16}{C}_6$$
(c) $${}^9{C}_3\cdot {}^8{C}_3$$
(d) $${}^{18}{C}_8$$

Answer 1

Hint:In this question, we first need to exclude a particular English book and a particular Telugu book from the given number of both the books. Then after removing the books we need to choose 6 books out of those remaining English and Telugu books which can be done using the formula $${}^n{C}_r={\displaystyle \frac{n!}{(n-r)!r!}}$$

Complete step-by-step answer:
COMBINATION: Each of the different selections which can be made by all of a number of given things without reference to the order of the things is called a combination
The number of combinations is given by the formula
$${}^n{C}_r={\displaystyle \frac{n!}{(n-r)!r!}}$$
Now, from the given 10 English books let us exclude a particular book
$$\Rightarrow 10-1$$
Now, on further simplification we have
$$\Rightarrow 9$$
Thus, we have 9 English books remaining after excluding a particular one
Let us now exclude a particular book from given 8 Telugu books
$$\Rightarrow 8-1$$
Now, this can be further written as
$$\Rightarrow 7$$
Thus, we have 7 Telugu books remaining after excluding a book.
Now, we have to choose 6 books out of these 9 English and 7 Telugu books
As we already know that we can choose the books using the formula of combinations
$${}^n{C}_r={\displaystyle \frac{n!}{(n-r)!r!}}$$
Let us first calculate how many books in total we have
$$\Rightarrow 9+7$$
Now, this can be written as
$$\Rightarrow 16$$
Now, let us choose 6 books out of these available 16 books which can be done as
$$\Rightarrow {}^{16}{C}_6$$
Hence, the correct option is (b).

Note:Instead of finding the number of each type of book after excluding we can directly calculate the total number of books we have and then subtract 2 books from that. Because there is no particular condition given in choosing the books. So, this process would not affect the result.It is important to note that before finding the number of possible ways of choosing we should exclude the particular book of each type because there is an option present in the question which can be chosen if we forget to exclude and it is incorrect.It is also to be noted that there is no particular condition given on choosing a particular book. So, we can add both of them and then choose 6.