Question

How many different selections of 4 books can be made from 10 different books, if
(i) there is no restriction;
(ii) two particular books are always selected;
(iii) two particular books are never selected?

Answer 1

(i) Required ways of selecting 4 books from 10 books without any restriction = ${}^{10}{C}_4=\frac{10}{4}\times \frac{9}{3}\times \frac{8}{2}\times 7=210$

(ii) Two particular books are selected from 10 books. So, 2 books need to be selected from 8 books.
Required number of ways if 2 particular books are always selected = ${}^8{C}_2=\frac{8}{2}\times \frac{7}{1}=28$

(iii) Two particulars books are never selected from 10 books. So, 4 books need to be selected from 8 books.
Required number of ways if two particular books are never selected = ${}^8{C}_4=\frac{8}{4}\times \frac{7}{3}\times \frac{6}{2}\times \frac{5}{1}=70$