PERMUTATIONS AND COMBINATIONS
Example 4 In how many ways can 5 children be arranged in a line such that (i) two
particular children of them are always together (ii) two particular children of them are
(i) We consider the arrangements by taking 2 particular children together as one
and hence the remaining 4 can be arranged in 4! = 24 ways. Again two particular
children taken together can be arranged in two ways. Therefore, there are
24 × 2 = 48 total ways of arrangement.
(ii) Among the 5! = 120 permutations of 5 children, there are 48 in which two children
are together. In the remaining 120 – 48 = 72 permutations, two particular children
are never together.
Example 5 If all permutations of the letters of the word AGAIN are arranged in the
order as in a dictionary. What is the 49th word?
Solution Starting with letter A, and arranging the other four letters, there are 4! = 24
words. These are the first 24 words. Then starting with G, and arranging A, A, I and N
in different ways, there are
12 words. Next the 37th word starts with I.
There are again 12 words starting with I. This accounts up to the 48 th word.
The 49th word is NAAGI.
Example 6 In how many ways 3 mathematics books, 4 history books, 3 chemistry
books and 2 biology books can be arranged on a shelf so that all books of the same
subjects are together.
Solution First we take books of a particular subject as one unit. Thus there are
4 units which can be arranged in 4! = 24 ways. Now in each of arrangements,
mathematics books can be arranged in 3! ways, history books in 4! ways,
chemistry books in 3! ways and biology books in 2! ways. Thus the total number
of ways = 4! × 3! × 4! × 3! × 2! = 41472.
Example 7 A student has to answer 10 questions, choosing atleast 4 from each of
Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways
can the student choose 10 questions?
Solution The possibilities are:
4 from Part A and 6 from Part B
5 from Part A and 5 from Part B
6 from Part A and 4 from Part B.
Therefore, the required number of ways is