1. A boy has nine trousers and 12 shirts. In how many different ways can he select a trouser and a shirt?
The boy can select one trouser in nine ways.
The boy can select one shirt in 12 ways.
The number of ways in which he can select one trouser and one shirt is 9 * 12 = 108 ways.
2. How many three letter words are formed using the letters of the word TIME?
The number of letters in the given word is four.
The number of three letter words that can be formed using these four letters is ⁴P₃ = 4 * 3 * 2 = 24.
3. Using all the letters of the word "THURSDAY", how many different words can be formed?
Total number of letters = 8
Using these letters the number of 8 letters words formed is ⁸P₈ = 8!.
4. 4. Using all the letters of the word "NOKIA", how many words can be formed, which begin with N and end with A?
There are five letters in the given word.
Consider 5 blanks ....
The first blank and last blank must be filled with N and A all the remaining three blanks can be filled with the remaining 3 letters in 3! ways.
The number of words = 3! = 6.
5. The number of arrangements that can be made with the letters of the word MEADOWS so that the vowels occupy the even places?
The word MEADOWS has 7 letters of which 3 are vowels.
As the vowels have to occupy even places, they can be arranged in the 3 even places in 3! i.e., 6 ways. While the consonants can be arranged among themselves in the remaining 4 places in 4! i.e., 24 ways.
Hence the total ways are 24 * 6 = 144.
6. The number of permutations of the letters of the word 'MESMERISE' is_.
a) 9!/(2!)^2 3!
b) 9!/(2!)^3 3!
c) 9!/(2!)^2 (3!)^2
d) 5!/(2!)^2 3!
n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.
The letter pattern 'MESMERISE' consists of 10 letters of which there are 2M's, 3E's, 2S's and 1I and 1R.
Number of arrangements = 9!/(2!)^2 3!
7. A committee has 5 men and 6 women. What are the number of ways of selecting 2 men and 3 women from the given committee?
The number of ways to select two men and three women = ⁵C₂ * ⁶C₃
= (5 *4 )/(2 * 1) * (6 * 5 * 4)/(3 * 2)
8. What are the number of ways to select 3 men and 2 women such that one man and one woman are always selected?
The number of ways to select three men and two women such that one man and one woman are always selected = Number of ways selecting two men and one woman from men and five women
= ⁴C₂ * ⁵C₁ = (4 * 3)/(2 * 1) * 5
= 30 ways.
9. A committee has 5 men and 6 women. What are the number of ways of selecting a group of eight persons?
Total number of persons in the committee = 5 + 6 = 11
Number of ways of selecting group of eight persons = ¹¹C₈ = ¹¹C₃ = (11 * 10 * 9)/(3 * 2) = 165 ways.
10. There are 18 stations between Hyderabad and Bangalore. How many second class tickets have to be printed, so that a passenger can travel from any station to any other station?
The total number of stations = 20
From 20 stations we have to choose any two stations and the direction of travel (i.e., Hyderabad to Bangalore is different from Bangalore to Hyderabad) in ²⁰P₂ ways.
²⁰P₂ = 20 * 19 = 380