Question 1 :
How many ways can the product a^{2}b^{3}c^{4} be expressed without exponents?
Solution :
From the given question, we come to know that "a" is appearing 2 times the letter "b" is appearing 3 times and the letter "c" is appearing 4 times.
Total number of letters = 2 + 3 + 4 = 9
Total number of ways = 9!/2!3!4!
= 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4!/(2 ⋅ 1) (3 ⋅ 2 ⋅ 1)4!
= 1260
Question 2 :
In how many ways 4 mathematics books, 3 physics books, 2 chemistry books and 1 biology book can be arranged on a shelf so that all books of the same subjects are together.
Solution :
Number of math books = 4
Number of physics books = 3
Number of chemistry books = 2
Number of biology book = 1
Here we have to consider set of each subject books as one units respectively.
So, we have 4 units of books
Number of arrangements in ordering math book = 4!
Number of arrangements in ordering physics book = 3!
Number of arrangements in ordering chemistry book = 2!
Number of arrangements in ordering biology book = 3!
Hence the total number of ways = 4! ⋅ 4! ⋅ 2! ⋅ 3!
= 6912
Question 3 :
In how many ways can the letters of the word SUCCESS be arranged so that all Ss are together?
Solution :
Let us consider all "S" together as one unit.
"C" is repeating 2 times, both "E" and "U" is repeating once.
Total number of letters = 5 (SSS as 1)
= 5!/(2!)
= 60
Hence the total number of words = 60.
Question 4 :
A coin is tossed 8 times,
(i) How many different sequences of heads and tails are possible?
(ii) How many different sequences containing six heads and two tails are possible?
Solution :
(i) Tossing 1 coin as 8 times and 8 coin as 1 time, both are equal.
So, the difference sequence of heads and tails = 2^{8}
(ii) Total number of entities in each entry = 8
Head is appearing 6 times, tail is appearing 2 times.
= 8!/(2! 6!)
= (8 ⋅ 7)/2
= 28
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